CHAPTER ONE: INTRODUCTION

1.1 Overview

This chapter describes a brief introduction to graph theory. History of graph theory and

some famous problems are introduced. Applications of graph labelings are listed and

explained to show the importance of a study in this area. Outline of this thesis with the

contents of each chapter is in this chapter as well.

1.2 Brief History of Graph Theory

A research area called graph theory was started by Leonhard Euler in 1736. The citizens of

Kaliningrad, Russia tried to solve if it was possible to cross all the bridges on the Pregel

river only once and arrive to the starting point. Euler as a well known mathematician used

graph representation and showed that it was impossible to do it.

Around 1850 another famous problem in graph theory was stated by Francis Guthrie. The

problem was called the Four Color Conjecture. In this problem, a map was to be colored in

such a way that all the adjacent countries sharing a border have different colors. The four

color theorem was proved using a computer by Appel and Haken in 1976. A non-computer

solution to the problem with an algorithm was not found until August 2004 when I. Cahit

[10] proposed a non-computer proof to the problem.

In the Four Color Conjecture the vertices which showed the countries are labeled as colors.

The vertices of a graph can be labeled in different ways such as labeling the vertices with

numbers. Labeling vertices and edges with numbers is a very basic and easy task. The task

is more complicated when we try to include other properties in the graph as well as labeling

the vertices and edges with different numbers.

The Magic-type labelings are thoroughly studied by Gallian [16]. It is stated in [16]:

“Motivated by the notation of magic squares in number theory, magic labelings were

1

introduced by Sedlaced in 1963.” Gallian summarized magic labelings, edge-magic total

labelings, super edge-magic labelings, vertex magic total labelings and summary of

antimagic labelings.

Vertex labeling methods with numbers include Vertex-Magic and Vertex-Antimagic

labeling. Vertex-Magic graphs are labeled in such a way that the sum of the vertex label

and its incident edge labels are same for every vertex. Vertex-Antimagic graphs are labeled

so that the sum of the vertex label and its incident edge labels are different for every vertex.

Computer-assisted proofs [11] are the subject of much controversy in the mathematical

world. Some mathematicians think that a computer-assisted proof is not a real mathematical

proof because they involve so many logical steps that they are not verifiable by human

beings, and that mathematicians are being asked to put their trust in computer

programming. A reverse question can also be raised; if computer calculations are not

trusted to carry out lengthy calculations, and since human beings are not infallible, why do

some researchers trust in lengthy human reasoning compared to machine computation?

Other mathematicians believe that computer-assisted proofs are as valid as any other type

of proof. The problem of human verifiability can be addressed by proving the computer

program itself valid. The computer-assisted proofs are subject to errors in their source

programs, compilers, and hardware, but this is resolved by multiple replications of the

result using different programming languages, different compilers, and different computer

hardware. In this thesis computer-assisted vertex-magic and vertex-antimagic total

labelings are found.

1.3 Applications of Graph Labelings

Graph labelings are becoming very useful models for some applications which include

coding theory problems such as the design of good radar-type codes, x-ray crystallography,

communication network addressing systems and circuit designs.

2

Gary Bloom and Solomon Golomb [8] have researched “real world” applications of

numbered undirected graphs. One of the applications to coding theory problem was to

design codes for pulse radar and missile guidance. This problem is defined as labeling a

graph with positive integers in such a way that the edges are different and the vertex labels

determine the time positions at which pulses are transmitted.

Another application of graph labeling is in communication addressing systems. Efficient

addressing systems is assigning addresses to the possible links in a communications

network where the addresses all have to be different and that the addresses of a link be

deduced from the identities of the two nodes linked, without having to use a lookup table.

The solution is as follows, first a graph of the network is constructed with nodes as vertices

and edges between all pairs of nodes where a link is provided. The vertices of the graph are

labeled in such a way that the differences between endpoints of edges are all different. Then

the address of a link is the difference between the labels on its endpoints. Suppose the

network graph is labeled with an edge-magic total labeling λ with a magic constant k. Then

the address of the link from x to Y is immediately calculated as .

Vertex-magic or vertex-antimagic labelings signify some additional information as well as

identifying vertices and edges. An increase in future applications of these labelings are

expected, therefore a research is done to find all possible labelings of path, cycle and tree

graphs in this area.

1.4 Outline of the Thesis

The aim of this thesis is to find all possible vertex-magic and (a,d)vertex-antimagic total

labelings for paths, cycles and some instances of trees. As discussed in section 1.3 the use

of graph labelings in network addressing, radar pulses and many other areas is critical.

Therefore a computer program is developed to find all possible vertex-magic and

(a,d)vertex-antimagic total labelings for some type of graphs. The restrictions and

boundaries of these graphs are discussed to set the domain of search area. An algorithm is

designed to find all possible labelings which are included in Appendix A and Appendix B.

3

Preliminaries to the thesis include a brief history of graph theory, the applications of graph

labelings and the outline of the thesis, in Chapter one.

In Chapter two, basic definition of graphs and basic knowledge of terms used in graph

theory is explained.

Chapter three includes a brief history of vertex-magic and vertex-antimagic labelings. Basic

counting on vertex-magic and vertex-antimagic labelings are also discussed in this chapter.

In Chapter four the problem to focus on in order to get all the possible results is explained.

Number of different possible labelings that is tried for vertex-magic and vertex-antimagic is

explained. The relation of the number of elements in a graph with the number of tries

needed to find all possible solutions is also discussed.

In Chapter five the solutions found are compared to previous proofs and open problems in

this area are studied.

1.5 Summary

Graph labelings have been used for a long time and they still have applications such as

producing good radar-type codes, x-ray crystallography, communication network

addressing systems and circuit designs. Magic labelings signify some more additional

property as well as labeling the elements of the graph. It is important to produce all possible

magic labelings of graphs for further use in applications listed above.

CHAPTER TWO: GENERAL KNOWLEDGE ON GRAPHS

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2.1 Overview

This chapter describes some basic definitions of graphs that will be used in this thesis.

Graph types and their properties are discussed. Vertex-magic and vertex-antimagic total

labelings are explained with their formal definitions.

2.2 Basic definitions

The basic definitions of graphs and its properties are discussed by Sugeng [46] as

explained; A graph is a finite set of vertices and edges where every edge connects two

vertices. A graph G consists of a finite set V(G) of elements called vertices and a set E(G)

of elements called edges. If x and Y are vertices in V(G) then the edge with endpoints x and

Y is indicated by the xy. A graph has order v and size e, where and .

A graph is finite if the order v is finite. A simple graph is of a kind that does not include

any edge with same endpoints. All graphs discussed in this thesis are finite and simple.

Graphs are composed of nodes and lines, where nodes are called vertices and lines are

called edges.

In a graph G, if vertex and there is an edge e between x and y, then x and Y are

called adjacent vertices. Vertex x is also called the neighbour of y. Moreover, both vertices

: Edge: Vertex

G1 G2 G3

Figure 2.1: Examples of simple finite graphs

u1 u2 u3

u4

v1

v2 v3

v4

v5v6

w1 w2 w3 w4

5

x and Y are incident with edge e. All of the neighbours of vertex x is denoted by

Degree of x is the number of neighbours of x. Therefore . If a vertex is

degree 0 then it is called an isolated vertex since it has no neighbours, and a vertex with

degree 1 is called an end vertex. The minimum degree of a graph G is

and the maximum degree is If

every vertex in the graph has the same degree r, , then G is a regular graph of

degree r, or an r-regular graph.

In graph G1 vertices u1, u3 and u4 are adjacent to u2. Therefore N(u2) = {u1, u3, u4} and

. In graph G2 since all the degrees of the vertices are the same and equal to 1,

then G2 is called a 1-regular graph. Graph G1 and G3 are not regular graphs. The vertex u2 in

G1 has degree 3 but the vertex u1 in the same graph has degree 1. Also the vertex w1 in G3

has degree 1 but w2 has degree 2.

Graph Y is called a subgraph of graph G if and . The graph G

is then called a supergraph of Y. A spanning subgraph Y is a subgraph of G such that

. In figure 2.2 (b), (c), (d), (e), (f) and (g) are all subgraphs of G4, but only

(b), (e), (f) and (g) are spanning subgraphs of G4.

A graph with n vertices x1,x2,…, xn and n-1 edges x1x2,x2x3,…,xn-1xn is called a path.

G4

Figure 2.2: Graph G4 and some of its subgraphs

u1 u2 u3

u4

(b)

u1 u2 u3

u4

(c)

u1 u3

u4

(d)

u1 u2

u4

(e)

u1 u2 u3

u4

(f)

u1 u2 u3

u4

(g)

u1 u2 u3

u4

6

A graph with n vertices x1,x2,…, xn and n edges x1x2,x2x3,…,xn-1xn,xnx1 is called a cycle.

A graph G is connected if for any two distinct vertices u and v of G there is a path between

u and v. Otherwise G is disconnected. A connected graph that does not contain a cycle is

called a tree. A path is a special kind of tree. Figure 2.3 gives examples of path P5 and cycle

C8.

A factor of a graph G is a spanning subgraph, a k-factor is a spanning k-regular subgraph.

Two graphs G1 and G2 of the same order are called isomorphic if there is a one-to-one

mapping f from G1 to G2 that keeps the adjacency property. Thus f(v1) is adjacent to f(v2) is

and only if v1 is adjacent to v2. If G1 = G2 then f is called an automorphism.

Two graphs G1 and G2 are called vertex disjoint graphs if Let G1 and

G2 be two vertex disjoint graphs. A union of G1 and G2, , is the graph that

consists of and

A complete graph Kn of order n is a graph in which every two distinct vertices are adjacent.

Let r be the degree of graph K, then r is, .

2.3 General Definitions of Vertex-Magic and Vertex-Antimagic Total Labelings

Figure 2.3: Path P5 and cycle C8

P5

v1

C8

u3 u4 u5

v2

v3

v4

v5v6

v7

v8

u2u1

7

Let G=(V,E) be a simple, finite and undirected graph with v vertices and e edges.

If graph G is labeled with numbers 1 through v + e such that every vertex and its incident

edges adds up to the same sum for every vertex, then this labeling is called a vertex-magic

total labeling of graph G. The identical sum in this graph is called the magic number. If

graph G is labeled with numbers 1 through v + e such that every vertex and its incident

edges add up to different sums for every vertex, then this labeling is called a vertex-

antimagic total labeling of graph G.

In both vertex-magic and vertex-antimagic total labelings, the sum of all labels associated

with a vertex is called the weight of that vertex. The weight of vertex , with labeling

α, is

In (a,d)-vertex antimagic labeling the smallest weighted vertex is a, and the other vertex

weights have a constant difference of d. A labeling α : V U E {1,2,…,n+e} is called a

(a,d)-vertex antimagic total labeling of G = G(V,E), if the set of vertex weights of all the

vertices in G is {a, a+d,…,a+(n-1)d} where and are fixed integers. If the

constant difference among weights is 0, d=0, then the labeling is called vertex magic total

labeling.

2.4 Summary

Graphs have some basic definitions such as vertex, edge, neighbor, subgraph etc. It is

required to learn these basics about graphs to understand studies in this research area. The

main focus of this thesis which is vertex-magic and vertex-antimagic total labelings are also

explained with their basic definitions.

CHAPTER 3: VERTEX-MAGIC AND VERTEX-ANTIMAGIC TOTAL LABELING

3.1 Overview

8

In this chapter’s beginning the history of magic labelings are discussed. The relation of

magic squares with magic labelings is described. Magic labelings are described, past

studies about magic labelings are listed and known results are stated in this chapter in

detail. Vertex-magic total labeling and vertex-antimagic total labeling is described in detail

with examples and basic counting is studied.

3.2 Brief History of Magic Labeling

3.2.1 Magic Squares

Magic squares are thoroughly explained by Wallis [52]; Magic squares are among the best

known mathematical recreations that have been known for ages. A magic square of side n is

an array whose entries are an arrangement of the integers {1, 2,…, n2}, in which all

elements in any row, any column, or either the main diagonal or main back-diagonal, add to

the same sum as in Figure 3.1.

1 15 8 10

12 6 13 3

14 4 11 5

7 9 2 16

Figure 3.1: Magic square with side = 4

Different entries to the square are also studied, such as all entries are primes or all entries

are perfect squares. Latin squares are studied since they are useful in constructing magic

squares. Magic rectangles are also an area of research which can be derived from Kotzig

arrays.

3.2.2 Magic Labeling

Wallis [52] explains that some authors introduced labelings that generalize the idea of a

magic square. Sedlacek defined a graph to be magic if it had an edge-labeling, with range

9

of real numbers, such that the sum of the labels around any vertex equals some constant,

independent of the choice of vertex.

Kotzig and Rosa defined a magic labeling to be a total labeling in which the labels are the

integers from 1 to . The sum of labels on an edge and its two endpoints is

constant. In 1996 Ringel and Llado redefined this type of labeling and called it edge-magic

labeling. Total labelings have also been studied in which the sum of the labels of all edges

adjacent to the vertex x, plus the label of x itself, is constant.

To clarify the terminological confusion described, we define a labeling to be vertex-magic

if the sum of all labels associated with a vertex equals a constant independent of the choice

of a vertex, and edge-magic if the same property holds for edges. The domain of the

labeling is specified by a modifier on the word “labeling”. For example, Stewart studies

vertex-magic edge labelings, and Kotzig and Rosa define edge-magic total labelings. This

thesis focuses on vertex-magic total labelings which is abbreviated to VMTL. The word

“total” is the modifier to the word “labeling”, in this kind of labelings all elements of the

graph (edges and vertices) are labeled.

As mentioned above Sedlacek [39] introduced magic labeling in 1963. Stewart [45] studied

on complete, basket and fan graphs to prove whether they can be labeled as magic graphs

or not. Stewart [44] also introduced semi-magic, where the labels of edges do not need to

start from 1. Jenzy and Trenkler [22] studied vertex magic edge labeling. Bodendiek and

Walther [9] introduced (a,d)-vertex-antimagic edge labeling (VAE). Baca [1] also studied

VAE labeling. Miller and Baca with some other researchers presented many results in

magic and antimagic labelings [1, 2, 4, 5, 34, 35]. Tezer and Cahit [50] studied on paths

and cycles for VAE.

Baca [3] introduced and studied (a,d)-vertex-antimagic total (VAT) labeling. Baca with

some researchers have done many studies on magic and antimagic labelings that also

includes VAT labelings [3,6]. MacQuillan[32] studied on various VAT labelings with

different properties.

10

MacDougall [30, 31] introduced and studied an instance of (a,d)-VAT labeling for d=0, and

he called it vertex magic total (VMT) labeling. Kovar [25,26] studied VMT labeling for

regular graphs as well as studying VAT labeling for cycles.

In their paper “Vertex-magic Total Labelings of Graphs” McDougall, Miller, Slamin,

Wallis [30] have studied some properties of these labelings and showed how to construct

labelings for several families of graphs, including cycles, paths, complete graphs of odd

order and the complete bipartite graph. They also showed that labelings are impossible for

some other classes of graphs. They have proven that; the n-cycle Cn has a labeling for any

. Pn, the path with n vertices, has a labeling for any . Every labeling of Pn is

derived from a labeling if Cn.

In the study “Vertex-Magic”, Daisy Cunningham [12] has studied on bounds on magic

numbers for cycles. Also, showed that if a graph has an odd number of vertices, algorithms

can be found to produce different vertex-magic graphs with the maximum and minimum

magic number. Cunningham has also given algorithms to produce a vertex-magic graph

with odd numbers or even numbers placed on the vertices for cycle graphs. In [12] the

following are also proved;

1. Let G be a cycle graph with v vertices where v is odd. There exists a vertex-magic

labeling with the numbers 1 to v located on the vertices and a magic number of ,the

upper bound for the magic number.

2. Let G be a cycle graph with v vertices where v is odd. There exists a vertex-magic

labeling with the numbers v+1 to 2v located on the vertices and a magic number of ,

the lower bound for the magic number

3. Let G be a cycle graph with v vertices where v is odd. There exists a vertex-magic

labeling for G with the odd numbers from 1 to 2v - 1 located on the vertices and a magic

number of 3v + 2.

11

4. Let G be a cycle graph with v vertices where v is odd. There exists a vertex-magic

labeling of G with the even numbers from 2 to 2v located on the vertices and a magic

number of 3v + 1

In [12] p.19 “Some interesting questions that arise from this paper are: For a given cycle

graph, is there a vertex-magic labeling associated with every magic number within the

bounds?” In this thesis this open problem written above is addressed with the complete list

of vertex-magic labelings produced by computer assistance.

In [3] “Vertex-antimagic Total Labelings of Graphs by Baca, Bertault, McDougall, Miller,

Simanjuntak and Slamin basic properties of (a,d)-vertex antimagic total labelings (VATL)

are studied. The relationships of VATL with several other previously studied graph

labelings are shown. Also showed how to construct labelings for certain families of graphs.

The following proofs are summarized from this paper;

1. Every odd cycle Cn, , has a -vertex-antimagic total labeling and a

-vertex-antimagic total labeling.

2. Every cycle Cn, , has a -vertex-antimagic total labeling and a -

vertex-antimagic total labeling.

3. Every cycle Cn, , has a -vertex-antimagic total labeling and a -

vertex-antimagic total labeling.

4. Every cycle Cn, , has a -vertex-antimagic total labeling and a -

vertex-antimagic total labeling.

5. Every odd cycle Cn, , has a -vertex-antimagic total labeling and a -

vertex-antimagic total labeling.

6. The path Pn has a (2n-1,1)-vertex-antimagic total labeling for any

One of the open problems for further research in [3] is as follows: “For the paths Pn and the

cycles Cn, determine if there is a vertex-antimagic total labeling for every feasible pair

(a,d).” In this thesis the open problem above is addressed with the complete list of

(a,d)-vertex antimagic labelings produced by computer assistance.

12

3.3 Description of Vertex-Magic Total Labeling

Vertex-magic total labeling is an assignment of the integers from 1 to to the vertices

and edges of graph G so that at each vertex the vertex label and the incident edge labels add

up to the same constant number.

In a more formal definition, vertex-magic total labeling is a one-to-one map λ from

onto the integers {1,2,…, } if there is a constant k so that for every vertex x,

λ(x) + Σ λ(xy) = k where the sum is on all vertices Y adjacent to x. The constant k in this

labeling λ is called the magic number (weight).

The notation of vertex-magic labeling was at least partially suggested by the following

question which appeared on a set of mathematical enrichment problems for high school

students:

The Olympic emblem consists of five overlapping rings containing 9 regions. In

order to contribute to a pension fund for a retiring IOC delegate, people are asked to

deposit money into each region. The guidelines allow the delegate to take all the

money in any one of the rings. Place $1, $2,…,$9 in the nine regions so that the

amount in each ring is the same.

8 3

6

9 1

7

4 2

5

Figure3.2: Solution to the Olympic rings problem

13

The Olympic rings problem can be defined as a vertex-magic labeling of a path with 5

vertices. Therefore the solutions to vertex-magic total labeling of P5 will give the answer to

the Olympic rings problem.

Weights of vertices of this graph is calculated,

wt(v1) = 8 + 3 = 11

wt(v2) = 7 + 3 + 1 = 11

wt(v3) = 6 + 1 + 4 = 11

wt(v4) = 5 + 4 + 2 = 11

wt(v5) = 9 + 2 = 11

As seen above all the weights are equal. The magic constant (k) in this labeling is 11. This

labeling is a vertex-magic total labeling.

An example of vertex-magic total labeling of a graph is in Figure 3.4. Each vertex and its

incident edges add up to 12. Every vertex-magic total labeling of a cycle graph with a

magic constant k can also be labeled as edge-magic graph with same magic constant as

shown in Figure 3.4. Every vertex-magic or edge-magic cycle graph can be changed to one

another by just shifting each label to the next element and maintaining the order of labels.

8 3 7 1 6 4 5 2 9

Figure3.3: Solution to the Olympic rings problem viewed in a path

v1 v2 v3 v4 v5e1 e2 e3 e4

14

Figure 3.4 shows just one way to create a vertex-magic graph with three vertices.

Depending upon the number of vertices and edges, a graph can be labeled in different ways

with different magic numbers.

3.3.1 Basic Counting on Vertex-Magic Total Labeling

As discussed in [30]; Let M = v + e and let Sv be the sum of all vertex labels and Se the sum

of all edge labels. Since the labels are from 1 to M, sum of all labels is

Sv + Se = =

(1)

In a labeling λ the magic constant k is calculated by

λ(x) + Σ λ(xy) = k

for only one vertex. When applied on all vertices, each vertex label is added only once and

each edge label is added twice to the sum, therefore

Sv + 2Se = vk (2)

When (1) and (2) is combined,

6

4 5

2

3

1

A vertex-magic cycle graph C3 with a magic number of 12

1

2 3

6

4

5

An edge-magic cycle graph C3 with a magic number of 12

Figure 3.4: Rotating the labels of a vertex-magic total label clockwise results in an edge-magic total label with same magic constant.

15

Se + = vk (3)

Vertices and edges are assigned distinct labels. Therefore the edges can receive the smallest

labels 1 to e, or the largest labels v+1 to M, or anything between these two maximum and

minimum points. Summarizing this we have,

(4)

A similar result also holds for Sv. Combining (3) and (4) gives us,

which will give us the range of feasible values for k. It is clear that when we know

and we can find the range of k.

3.4 Description of Vertex-Antimagic Total Labeling

Vertex-antimagic total labeling is an assignment of the integers from 1 to v+e to the

vertices and edges of G so that at each vertex the vertex label and the incident edge labels

add up to the different numbers.

In a more formal definition, vertex-antimagic total labeling is a one-to-one map λ from

E U V onto the integers {1,2,…,v+e} if the weights of vertices wt(x), x Element of V are

pairwise distinct.

V U E = {1, 2,…, v + e} is called an (a,d)-vertex antimagic total labeling (VATL) of graph

G if the set of vertex weights is W = { wt{x}|x ELEMENT V} = {a, a + d,…, a + (v-1)d}

for some integers a and d.

Figure 3.5 shows a vertex-antimagic total labeling and (a,d)-vertex-antimagic labeling of

graph G. Let V(G) = {2, 5, 6, 8, 9} and E(G) = {1, 3, 4, 7, 10}.

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3.4.1 Basic Counting on Vertex-Antimagic Total Labeling

As discussed in [3]; Let M = v + e and let Sv be the sum of all vertex labels and Se the sum

of all edge labels. Since the labels are from 1 to M, sum of all labels is

Sv + Se = =

(1)

Let , when summed on all vertices, each vertex label is added only once and

each edge label is added twice, therefore

Sv + 2Se = (2)

Combining (1) and (2) gives us,

Se + =

(3)

8 5

6

2

9

7

10 4

31

wt(V1) = 2 + 1 + 3 = 6wt(V2) = 6 + 3 + 4 = 13wt(V3) = 5 + 4 + 7 = 16wt(V4) = 8 + 7 + 10 = 25wt(V5) = 9 + 10 + 1 = 20

2 8

9

6

5

10

7 4

31

wt(V1) = 6 + 1 + 3 = 10wt(V2) = 9 + 3 + 4 = 16wt(V3) = 8 + 4 + 10 = 22wt(V4) = 2 + 10 + 7 = 19wt(V5) = 5 + 7 + 1 = 13

Vertex-Antimagic Label (10,3)-Vertex-Antimagic Label

Figure 3.5: A Vertex-Antimagic and a (a,d)Vertex-Antimagic Total Label

17

The edge labels can receive the e smallest labels or e largest labels or anything between.

Therefore we have,

(4)

A similar result also holds for Sv. Combining (3) and (4),

Shows the feasible values of a and d that are restricted. It is possible to get stronger

restrictions for particular graphs.

Let δ be the smallest degree in graph G, then the minimum possible weight on a vertex is at

least 1 + 2 + … + (δ + 1), therefore

(5)

Similarly if φ is the largest degree, then the maximum vertex weight is no more than the

sum of φ + 1 largest labels. Therefore,

(6)

Combining the inequalitites (5) and (6) the upper bound on value of d is found and shown

as follows:

(7)

3.5 Summary

Magic squares is one of the well known mathematical problems and magic labelings are

introduced to generalize the idea of magic squares. Magic labelings are studied on since

1963. Vertex-magic and vertex-antimagic total labelings are discussed in detail with

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previously studied basic counting. It is required to study known results and previously

studied basic countings to produce a good algorithm.

CHAPTER 4: ALGORITHM ON VERTEX-MAGIC AND VERTEX-ANTIMAGIC

TOTAL LABELING

4.1 Overview

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The problem is to find all possible vertex-magic total labelings(VMTL) and (a,d)vertex-

antimagic total labelings(VATL) of cycles and paths. A computer program is written in C

language to try all different possible labelings of cycles and paths. Trying all the possible

labelings on a graph will give all the possible solutions of vertex-magic and vertex-

antimagic total labelings.

The computer programs written are categorized in two main groups. One program is

written to find VMTL and the other is for (a,d)-VATL. These programs and their

algorithms are discussed in this chapter.

4.2 Program for Vertex-Magic Total Labelings

In this section the programs for cycle, path and tree graphs are observed. The problem is

explained and the restrictions used in the program are discussed.

4.2.1 Vertex-Magic Total Labeling of Cycles

A program is written to find all possible vertex-magic total labelings on cycle graphs with

three, four, five, six, seven and eight vertices and edges. Figure 4.1 below shows a cycle

with v,e = 4, C4.

Numbers from 1 to v + e = 8 are to be labeled on Figure 4.1 to find all possible vertex-

magic total labelings. All possible different labelings are labeled and checked to see if it is

a vertex-magic labeling or not.

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For a graph with 8 elements the number of distinct sets of V(G) and E(G) can be calculated

by using combinations;

Therefore, there are 70 different possibilities with different elements in each set in each

instance. Table 4.1 shows all different combinations of labels for the cycle graph with v,e =

4, C4.

v1 v2

v3v4

e4

e1

e2

e3

Figure 4.1: Cycle, C4, with four vertices and edges

No Edges Vertices No Edges Vertices No Edges Vertices

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

1,2,3,4

1,2,3,5

1,2,3,6

1,2,3,7

1,2,3,8

1,2,4,5

1,2,4,6

1,2,4,7

1,2,4,8

1,2,5,6

1,2,5,7

1,2,5,8

1,2,6,7

1,2,6,8

1,2,7,8

1,3,4,5

1,3,4,6

1,3,4,7

1,3,4,8

1,3,5,6

1,3,5,7

1,3,5,8

1,3,6,7

1,3,6,8

5,6,7,8

4,6,7,8

4,5,7,8

4,5,6,8

4,5,6,7

3,6,7,8

3,5,7,8

3,5,6,8

3,5,6,7

3,4,7,8

3,4,6,8

3,4,6,7

3,4,5,8

3,4,5,7

3,4,5,6

2,6,7,8

2,5,7,8

2,5,6,8

2,5,6,7

2,4,7,8

2,4,6,8

2,4,6,7

2,4,5,8

2,4,5,7

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

1,3,7,8

1,4,5,6

1,4,5,7

1,4,5,8

1,4,6,7

1,4,6,8

1,4,7,8

1,5,6,7

1,5,6,8

1,5,7,8

1,6,7,8

2,3,4,5

2,3,4,6

2,3,4,7

2,3,4,8

2,3,5,6

2,3,5,7

2,3,5,8

2,3,6,7

2,3,6,8

2,3,7,8

2,4,5,6

2,4,5,7

2,4,5,8

2,4,5,6

2,3,7,8

2,3,6,8

2,3,6,7

2,3,5,8

2,3,5,7

2,3,5,6

2,3,4,8

2,3,4,7

2,3,4,6

2,3,4,5

1,6,7,8

1,5,7,8

1,5,6,8

1,5,6,7

1,4,7,8

1,4,6,8

1,4,6,7

1,4,5,8

1,4,5,7

1,4,5,6

1,3,7,8

1,3,6,8

1,3,6,7

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

2,4,6,7

2,4,6,8

2,4,7,8

2,5,6,7

2,5,6,8

2,5,7,8

2,6,7,8

3,4,5,6

3,4,5,7

3,4,5,8

3,4,6,7

3,4,6,8

3,4,7,8

3,5,6,7

3,5,6,8

3,5,7,8

3,6,7,8

4,5,6,7

4,5,6,8

4,5,7,8

4,6,7,8

5,6,7,8

1,3,5,8

1,3,5,7

1,3,5,6

1,3,4,8

1,3,4,7

1,3,4,6

1,3,4,5

1,2,7,8

1,2,6,8

1,2,6,7

1,2,5,8

1,2,5,7

1,2,5,6

1,2,4,8

1,2,4,7

1,2,4,6

1,2,4,5

1,2,3,8

1,2,3,7

1,2,3,6

1,2,3,5

1,2,3,421

Table 4.1: Possible distinct sets of Edges and Vertices in C4

The cycle, C4, is labeled using labeling number 3 from Table 4.1 as shown in Figure 4.2.

Realize that the weights are all different. Therefore this labeling is not a vertex-magic total

labeling. After a careful observation, it is realized that by just changing the position of

labels in either set will result in different weights on the vertices. Without changing the

elements of V and E and by just changing the positions of “1” and “3” in set E, we get

Figure 4.3.

Set V(G) has four elements. It is possible to place those four elements in different positions

and each position results in a different labeling. The amount of possible positions that four

elements have is calculated by using permutations [38]:

V(G) has twenty four different positions with a same set of elements. The Table 4.2 shows

all the different positions of labeling number 3 from Table 4.1 with a fix positioning on

E(G) and changing positions of V(G).

1 5

2

738

6

4Calculating the weights on each vertex,wt(v1) = 4 + 6 + 1 = 11wt(v2) = 5 + 1 + 2 = 8wt(v3) = 7 + 2 + 3 = 12wt(v4) = 8 + 3 + 6 = 17

Figure 4.2: A labeling of C4 with calculated weights

3 5

2

718

6

4Calculating the weights on each vertex,wt(v1) = 4 + 6 + 3 = 13wt(v2) = 5 + 3 + 2 = 10wt(v3) = 7 + 2 + 1 = 10wt(v4) = 8 + 1 + 6 = 15

Figure 4.3: A different labeling of C4 with changed positions in set E

22

No Edges Vertices No Edges Vertices No Edges Vertices

1

2

3

4

5

6

7

8

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

4,5,7,8

4,5,8,7

4,7,5,8

4,7,8,5

4,8,5,7

4,8,7,5

5,4,7,8

5,4,8,7

9

10

11

12

13

14

15

16

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

5,7,4,8

5,7,8,4

5,8,4,7

5,8,7,4

7,4,5,8

7,4,8,5

7,5,4,8

7,5,8,4

17

18

19

20

21

22

23

24

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

1,2,3,6

7,8,4,5

7,8,5,4

8,4,5,7

8,4,7,5

8,5,4,7

8,5,7,4

8,7,4,5

8,7,5,4

Table 4.2: Elements of set V positioned in 24 different ways to produce different labelings

Labelings in Table 4.2 are drawn in Figure 4.5. Figure 4.4 shows the places of labels as a

reference guide to Figure 4.5. The weights of vertices are calculated and labeled inside the

gray area. The edges and vertices are labeled to the places seen in Figure 4.4.

wt(V1)

V1

wt(V4)

wt(V2)

wt(V3)

V2

V3V4

E1

E2

E3

E4

Edges = {E1,E2,E3,E4}

Vertices = {V1,V2,V3,V4}

Figure 4.4: Representation of edges and vertices onto a graph

23

Looking at the twenty four labelings, we realize that none of them are VMTL. The question

“Have we tried all the possible labelings for graph G with V(G) = {1,2,3,6} E(G) =

873

4 58

1

26

1316

11

583

4 710

1

26

1017

11

853

4 710

1

26

1314

11

573

4 811

1

26

1116

11

753

4 811

1

26

1214

11

783

5 47

1

26

1217

12

873

5 47

1

26

1316

12

483

5 710

1

26

917

12

843

5 710

1

26

1313

12

473

5 811

1

26

916

12

743

5 811

1

26

1213

12

583

7 47

1

26

1017

14

853

7 47

1

26

1314

14

483

7 58

1

26

917

14

843

7 58

1

26

1313

14

2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

783

4 58

1

26

1217

11

1

953

7 811

1

26

1514

14

1043

7 811

1

26

1613

14

573

8 47

1

26

1016

15

753

8 47

1

26

1214

15

473

8 58

1

26

916

15

743

8 58

1

26

1213

15

453

8 710

1

26

914

15

543

8 710

1

26

1013

15

17 18 19 20

21 22 23 24

Figure 4.5: Drawn Figures of labelings in Table 4.2

24

{4,5,7,8} ?” comes to mind immediately. The answer is no. For each permutation of V(G),

E(G) also has to be permuted. For each different labelings of V(G), all twenty four

permutations of E(G) has to be tried.

Keep the set V(G)={5,8,7,4} in a fixed position and permute E(G). Table 4.3 shows all

different positioning.

No Edges Vertices No Edges Vertices No Edges Vertices

1

2

3

4

5

6

7

8

1,2,3,6

1,2,6,3

1,3,2,6

1,3,6,2

1,6,2,3

1,6,3,2

2,1,3,6

2,1,6,3

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

9

10

11

12

13

14

15

16

2,3,1,6

2,3,6,1

2,6,1,3

2,6,3,1

3,1,2,6

3,1,6,2

3,2,1,6

3,2,6,1

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

17

18

19

20

21

22

23

24

3,6,1,2

3,6,2,1

6,1,2,3

6,1,3,2

6,2,1,3

6,2,3,1

6,3,1,2

6,3,2,1

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

5,8,7,4

Table 4.3: Elements of Set E positioned in 24 different ways to produce different labelings

Looking at 3rd labeling in Table 4.3, we realize a vertex magic total labeling is obtained

shown in Figure 4.6.

Therefore, since V(G) with four elements have 24 possible different positions and E(G)

with four elements have 24 different positions different possible labelings

exists for each distinct set of V(G) and E(G) in a cycle graph, C4. Recall from Table 4.1 that

there are a total of 70 different labelings in which V(G) and E(G) have distinct elements.

Therefore,

25

742

5 812

1

36

1212

12

Figure 4.6: A vertex-magic total label of C4

A total of 40,320 different labelings are labeled on the cycle graph G with v,e = 4. At each

instance the weights of the vertices are calculated and checked if they are all the same or

not. If the weights are all same, the vertex magic total label is displayed on the screen. This

process is summarized in Figure 4.7.

The cycle graph, C4, with four vertices and edges is discussed. The number of labelings and

the total calculations calculated for cycles, Cn, with is given in Table 4.4. Total

calculations is calculated by,

START

END

LABELINGS (70times):Determine V(G) and E(G)

PERMUTE V(G) (24times):V(G) is set in a different order with same elements

PERMUTE E(G) (24times):E(G) is set in a different order with same elements

Calculate the weights

Check if all the weights are the same

Display the solution

YES

Figure 4.7: Flowchart of the program to search C4

26

Cycle Distinct Labels Total variations

of V(G)

Total variations

of E(G)

Total

Calculations

C3 720

C4 40,320

C5 3,628,800

C6 479,001,600

C7 87,178,291,200

C8 20,922,789,888,000

Table 4.4: Number of labelings, number of variations of V and E, and total calculations of cycles

The program for cycle graph G with eight vertices and edges, C8, is written but it is not

completed due to lack of computer power. C7 with 87,178,291,200 calculations is

completed in approximately 8.95hours. Approximate completion time of C8 is

times more calculations which results in approximately

to complete. Future plans include separating the

program into parts and run them over multiple computers for a result on C8.

4.2.2 Vertex-Magic Total Labeling of Paths

Let Pn be a path with n vertices. A computer program to find all possible vertex-magic total

labelings of Pn where is written. In the figure below a path P4 is shown

Realize that a path with 4 vertices only has 3 edges. Realize when calculating the weights

on a path, the first and last vertices are incident to only one edge. Therefore the calculation

of the weights of Figure 4.8 is,

wt(V1) = V1 + E1

V1 V2 V3 V4

E1 E2 E3

Figure 4.8: A path, P4, with four vertices and three edges

27

wt(V2) = V2 + E1 + E2

wt(V3) = V3 + E2 + E3

wt(V2) = V4 + E3

In this graph V(G) has 4 elements and E(G) has 3 elements. The target is to check all the

cases of how this path can be labeled. Label the graph G, , with numbers starting

with 1 to v + e. Therefore a calculation using combinations is done to find the amount of

distinct sets of E(G) that is available.

Thirty five different labelings of E(G) is possible and they are shown in Table 4.5.

Labeling

number

E V Labeling

number

E V

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

1,2,3

1,2,4

1,2,5

1,2,6

1,2,7

1,3,4

1,3,5

1,3,6

1,3,7

1,4,5

1,4,6

1,4,7

1,5,6

1,5,7

1,6,7

2,3,4

2,3,5

2,3,6

4,5,6,7

3,5,6,7

3,4,6,7

3,4,5,7

3,4,5,6

2,5,6,7

2,4,6,7

2,4,5,7

2,4,5,6

2,3,6,7

2,3,5,7

2,3,5,6

2,3,4,7

2,3,4,6

2,3,4,5

1,5,6,7

1,4,6,7

1,4,5,7

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

2,3,7

2,4,5

2,4,6

2,4,7

2,5,6

2,5,7

2,6,7

3,4,5

3,4,6

3,4,7

3,5,6

3,5,7

3,6,7

4,5,6

4,5,7

4,6,7

5,6,7

1,4,5,6

1,3,6,7

1,3,5,7

1,3,5,6

1,3,4,7

1,3,4,6

1,3,4,5

1,2,6,7

1,2,5,7

1,2,5,6

1,2,4,7

1,2,4,6

1,2,4,5

1,2,3,7

1,2,3,6

1,2,3,5

1,2,3,4

Table 4.5: Possible distinct sets of edges and vertices in P4

28

Labeling number 12 in Table 4.5 of path P4 with E(G) = {1,4,7} and V(G) = {2,3,5,6} is

drawn in Figure 4.9.

Calculating the weights,

wt(V1) = 2 + 1 = 3

wt(V2) = 3 + 1 + 4 = 8

wt(V3) = 5 + 4 + 7 = 16

wt(V4) = 6 + 7 = 13

The weights are all different. Therefore this labeling is not a vertex-magic total labeling.

The previous study has shown that different positions with the same elements on V(G) and

E(G) results in different results. Keep the E(G) in a fixed position and try different

positions on elements of V(G). There are,

different variations of V(G) with the same elements as shown in Table 4.6.No E(G) fixed V(G) No E(G)

fixed

V(G)

1

2

3

4

5

6

7

8

9

10

11

12

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

2,3,5,6

2,3,6,5

2,5,3,6

2,5,6,3

2,6,3,5

2,6,5,3

3,2,5,6

3,2,6,5

3,5,2,6

3,5,6,2

3,6,2,5

3,6,5,2

13

14

15

16

17

18

19

20

21

22

23

24

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

1,4,7

5,2,3,6

5,2,6,3

5,3,2,6

5,3,6,2

5,6,2,3

5,6,3,2

6,2,3,5

6,2,5,3

6,3,2,5

6,3,5,2

6,5,2,3

6,5,3,2

Table 4.6: Elements of set V positioned in 24 different ways to produce different labelings for P4

2 3 5 6

1 4 7

Figure 4.9: A labelled path P4

29

Assume that all 24 different labelings are calculated. None of them are vertex-magic total

labelings. Lets focus on labeling number 23. This labeling is shown in Figure 4.10.

The weights are not same, therefore this is not a vertex-magic total labeling. From the study

in previous chapter we know that by only permuting V(G) or E(G) a result is not always

obtained, so try permuting both to try all possibilities. In labeling number 23 shown in

Figure 4.10 above, change the position of elements of E(G) from {1,4,7} to {4,1,7}. The

new labeling that is formed is shown in Figure 4.11.

The weights are all the same. This is a vertex-magic total labeling with a magic constant

k=10. This shows that it is not enough to keep the elements of E(G) in fixed positions and

permute the elements of V(G) only. The permutation number for V(G) is calculated in (4)-

above formula. The number of permutations of E(G) is

In a path P4, for each label there is 6 different variations of E(G) and 24 different variations

of V(G) with a total of

different variations. From previous calculations it is found that there are 35 distinct

labelings. Therefore,

6 5 2 3

1 4 7Calculating the weights,

wt(V1) = 6 + 1 = 7

wt(V2) = 5 + 1 + 4 = 10

wt(V3) = 2 + 4 + 7 = 13

wt(V4) = 3 + 7 = 10Figure 4.10: Path P4 labelled with calculated weights

6 5 2 3

4 1 7Calculating the weights,

wt(V1) = 6 + 4 = 10

wt(V2) = 5 + 4 + 1 = 10

wt(V3) = 2 + 1 + 7 = 10

wt(V4) = 3 + 7 = 10Figure 4.11: Path P4 labelled with different positions of edge labels

30

calculations are done in order to get all possible labelings to this particular graph.

In this section a path P4 is discussed. The number of labelings available and number of

calculations required for Pn, where is shown in Table 4.7. Total calculations are

calculated as follows;

E V Labelings Total Variations

of E(G)

Total Variations

of V(G)

Total Calculations

3 4 5,040

4 5 362,880

5 6 39,916,800

6 7 6,227,020,800

7 8 1,307,674,368,000

8 9 355,687,428,096,000

Table 4.7: Number of labelings, number of variations of V and E, and total calculations of Paths

P7 is completed in 1.3hours. To complete P8 times more

computer power is needed compared to P7. Approximately is needed

to complete P8 which is not completed in this thesis. P9 is also not completed due to

extreme computer power requirements.

4.2.3 Vertex-Magic Total Labeling of Binary Tree

In [49], the binary tree with seven vertices and six edges in Figure 4.12 is proved that it has

no possible VMT labeling. A conjecture “Complete Binary Tp has no VMT labeling” is also

31

stated in [49]. A computer program is developed to check if the tree in Figure 4.12 has any

VMT labeling.

In this computer program, the number of possible labelings and total calculations

completed is shown below. The number of different sets of E(G) and V(G) is,

Different possible variations of each set is calculated,

Therefore the number of possible labelings that is calculated is

.

On a computer with a Pentium 4 2.8Ghz processor, the result shown below is obtained;

Number of tries: 6.22702e+09

Number of solutions: 0

Number of labels: 1716

The time taken was: 1771.3 seconds

The time taken is given in seconds, which means it took approximately 30 minutes to do

6,227,020,800 tries.This result shows that the tree in Figure 4.12, has no vertex-magic total

labeling.

V1

V2

V3 V4

V5

V6 V7

E1 E2

E3 E4 E5 E6

Calculation of weights for binary tree with seven vertices is as follows,

wt(V1)=V1 + E1 + E2 wt(V2)=V2 + E1 + E3 + E4

wt(V3)=V3 + E3

wt(V4)=V4 + E4

wt(V5)=V5 + E2 + E5 + E6

wt(V6)=V6 + E5

wt(V7)=V7 + E6

Figure 4.12: Binary tree with seven vertices and six edges

32

After this result was obtained, we tried to see if there is any vertex magic total labeling if

another edge and vertex is added to the tree. Therefore the tree in Figure 4.13 below is

searched for a vertex magic total labeling.

To find out the number of possible labelings and total calculations is calculated below.

The number of distinct sets with different elements in E(G) and V(G) is,

Then the different possible variations of each set is calculated,

Therefore the number of possible variations that is calculated is

Recall from the previous calculation for Figure 4.12 that 6,227,020,800 tries is completed

in 1771seconds. An approximation of the time needed for 1,307,674,368,000 tries is

calculated as,

In order to complete this research, a computer laboratory with seventeen computers was

used for one day. The computer program that searches is composed of loops to try all

possibilities. This program is divided equally into seventeen smaller programs with

V1

V2

V3 V4

V5

V6 V7

E1 E2

E3 E4 E5 E6

Calculation of weights of seven vertices is as follows,

wt(V1)=V1 + E1 + E2 wt(V2)=V2 + E1 + E3 + E4

wt(V3)=V3 + E3 + E7

wt(V4)=V4 + E4

wt(V5)=V5 + E2 + E5 + E6

wt(V6)=V6 + E5

wt(V7)=V7 + E6

wt(V8)=V8 + E7

V8

E7

Figure 4.13: Binary tree with an added edge and vertex

33

diminished loops. Each smaller program completes different possibilities of labelings. After

all programs are completed, all possibilities were tried. Each smaller program was loaded

into a computer and executed. All of the smaller programs completed successfully. The

results on each computer are combined together. All twelve vertex magic total labelings are

in Appendix A.

Adding another vertex and an edge to the tree is considered and the time required to

complete such a tree is calculated,

The number of different sets of V(G) and E(G) is calculated as,

Then the different possible variations of each set is calculated,

Therefore the number of total tries (labelings) that is required to be calculated is

To complete 355,687,428,096,000 tries, with the knowledge that 1,307,674,368,000 tries

takes approximately 103 hours,

Realizing the extreme requirements for such a research, this tree is not studied in this thesis.

Therefore the conjecture in [49] is not completely answered due to lack of computer power,

but the results found for Figure 4.13 shows that there might be a VMT labeling for a

complete binary tree.

4.3 Program for (a,d)Vertex-Antimagic Total Labelings

34

A program is run to find (a,d)vertex-antimagic total labelings for cycles and paths. The

program of (a,d)vertex-antimagic labelings are categorized in two main categories. One of

them is designed for cycles and the other one for paths.

4.3.1 (a,d)Vertex-Antimagic Total Labeling for Cycles

In a (a,d)vertex-antimagic total labeling numbers 1 to v + e is labeled onto graph elements

so that a set of vertex weights W = {a, a + d,…, a + (v-1)d} is searched for where a is the

starting weight, d is the increment, v is the number of vertices and e is the number of edges.

A vertex-antimagic total labeling is a graph labeling in which all the weights are different.

In Figure 4.14 a cycle graph, C5, with five vertices and edges is labeled from 1 to

v + e = 10.

Randomly placed labels formed a vertex-antimagic total labeling, since all of the weights

are different. A more challenging labeling is to find a (a,d)-vertex-antimagic total labeling.

To answer this question one has to look at the weights and search for a arithmetic

progression. Looking at Figure 4.14 and calculating weights of each vertex the set W =

{13,11,17,23,21} is obtained. By sorting the weights in an ascending order we get W =

{11,13,17,21,23}. Look at how the weights are increasing, if a constant increment is found

then this is a (a,d)vertex-magic total labeling. There is an increment of 2, then a 4,4,2

consecutively. The increment is not constant. Therefore this labeling is not a (a,d)-vertex-

9 7

5

1

3

6

8 4

210 wt(V1) = 1 + 10 + 2 = 13wt(V2) = 5 + 2 + 4 = 11wt(V3) = 7 + 4 + 6 = 17wt(V4) = 9 + 6 + 8 = 23wt(V5) = 3 + 8 + 10 = 21

Figure 4.14: A vertex-antimagic total labeling of cycle graph C5

35

antimagic labeling. Look at a different labeling of cycle C5 and observe that there is a

constant increment as shown below

In Figure 4.15 the set of weights is W = {6, 11, 21, 26, 16}. Order this set in an ascending

order to see the increment more clearly, the set W = {6, 11, 16, 21, 26} has a constant

increment k = 5. This labeling is a (6,5)-vertex-antimagic labeling for C5.

The minimum weight that can be obtained in a labeling V U E = {1, 2, 3,…, 2v} is the

addition of three smallest labelings, as shown in Figure 4.16.

The maximum weight that can be obtained in a labeling V U E = {1, 2, 3,…,2v} is the

addition of three largest labelings, as shown in Figure 4.16.

It is very important to understand the maximum and minimum weights that can occur in a

graph. It is not productive to search a labeling for instances that are not possible. Such as

9 6

4

3

8

10

7 5

21 wt(V1) = 3 + 1 + 2 = 6wt(V2) = 4 + 2 + 5 = 11wt(V3) = 6 + 5 + 10 = 21wt(V4) = 9 + 10 + 7 = 26wt(V5) = 8 + 7 + 1 = 16

Figure 4.15: A (6,5)vertex-antimagic total labeling of cycle graph C5

2

1

3

6

2v

2v-1

2v-2

6v-3

Figure 4.16: Maximum and minimum possible weights

36

searching for an instance where a is below the minimum possible weight, or a weight that is

above the maximum possible weight.

In C5 the minimum weight is obtained using (5) in chapter 3.3.1,

, where is the smallest degree in a graph. For C5 =2. Therefore,

. Thus

And the maximum is obtained using (5) in chapter 3.3.1,

“Similarly if φ is the largest degree, then the maximum vertex weight is no more than the

sum of φ + 1 largest labels”

For C5 φ=2. Therefore, sum of largest φ + 1 = 2 + 1 = 3 labels gives us maximum vertex

weight. Sum of largest 3 labels for C5 is 10 + 9 + 8 = 27.

Therefore there is no need to search for labelings of a < 6 since a cannot be less than 6.

Since the minimum increment is 1 and if a is larger than 23, the set of weights W(G) will

have an element which exceeds the maximum possible weight. Assume starting weight, a,

is 24, even with a minimum increment of 1, the set will include {24, 25, 26, 27, 28}. As we

have discussed above the maximum weight that can be obtained is 27. Therefore it is

unnecessary to search a labeling with a starting weight less than 6 or more than 23 for C5.

for C5

The domain of d is also carefully studied to avoid unnecessary calculations. The increment

on the progression is an integer with a minimum value of 1. There should be a limitation to

the increment value at the upper bound. As stated above, the minimum weight a label can

have is 6. Assuming that in a labeling a is 6 and d is 5. The elements of W = {6, 11, 16, 21,

26}. The maximum weight in the set is 26 and it can be obtained in C5. Another search on

labelings for C5 with a = 6 and d = 6 will result in a set like this W = {6, 12, 18, 24, 30}.

In this case the set contains a weight which is beyond a possible reach, therefore d=6

shouldn’t be tried. Boundaries of the increment constant, d, is calculated from (7) in chapter

3.3.1

37

For C5, M = v+e = 5+5 = 10, largest degree, = 2, smallest degree, = 2 and v = 5

In the figure below C5 is labeled in different (a,d)-vertex-antimagic total labelings.

Figure 4.17 shows only few possible (a,d)-vertex antimagic total labelings for C5. All of the

results for cycles C3, C4, C5, C6, C7 and C8 are in Appendix B.

9

16

6

4

3

8

10

7 5

216

11

2126

(6,5)vertex-antimagicThe minimum weight 6 is obtained with a maximum increment

6

16

7

3

9

5

2

10 8

4114

15

1718

(14,1)vertex-antimagicThe minimum increment is used.

8

17

7

3

1

6

10

9 5

427

12

2227

(7,5)vertex-antimagicThe maximum weight 27 is obtained with a maximum increment

2

15

6

3

4

5

10

9 8

7112

18

2421

(12,3)vertex-antimagic

Figure 4.17: Examples of (a,d)-vertex antimagic total labelings for C5

38

Table 4.8 shows the feasible range of a and d in cycle graphs, Cn, where

Cycle a d Smallest weight Largest weight

C3 6 15

C4 6 21

C5 6 27

C6 6 33

C7 6 39

C8 6 45

Table 4.8: Search range of a and d, smallest and largest possible weights for cycles

As discussed before to complete the systematic search of C8 approximately 89.5days of

computer power is required. Therefore C8 is not completely searched, except the instance of

a=6 and d=5. Using the formula (2) from section 3.3.1 it is possible to eliminate most of the

labelings that are not feasible to produce a (6,5)vertex-antimagic total labeling.

Sv + 2Se = , substituting a=6, d=5, v=4 into the formula

Sv + 2Se = Sv + 2Se = 54

Since Sv and Se are calculated in all labelings, only the labelings that satisfies this equality

are permuted to find a (6,5)vertex-antimagic total labeling. This criteria decreases the

amount of labelings to be tried, which decreases the computer power required and makes it

possible to be solved.

Recall from section 4.2.2 that a cycle graph C4 has 70 distinct labelings. Each distinct

labeling has different sets of V(G) and E(G). In each labeling all of the permutations of

V(G) and E(G) are labeled to the graph to check if it is a vertex-magic total labeling. The

weights are compared and checked if they are all equal, as the condition below

39

C4 with 70 distinct labelings, 24 different orders of V(G) and 24 different orders of E(G)

results in

different labelings. All 40,320 labelings are tested for one condition only. If one logical test

is referred as 1 job or calculation, then the computer has to do

calculations in order to finish the desired work.

It is not the same in (a,d)-vertex-antimagic labelings. For each different labeling of C4 the

computer needs to apply the following logical tests to find the desired solutions,

The variable start will have 13 different values and increment will have 5 different values.

Therefore for each different labeling to check for all possible progressions in a C4 cycle, the

computer checks for

In (a,d)vertex-antimagic total labeling, for each label are

checked compared to vertex-magic labeling. In a total of 40,320 labels

extra conditions are checked. An increase in computation time is

observed when searching for a (a,d)vertex-antimagic total labeling compared the

computation time when searching for vertex-magic total labeling.

4.3.2 (a,d)Vertex-Antimagic Labeling for Paths

A program for a path Pn with n vertices is written to find all possible (a,d)-vertex-antimagic

labelings where . The search domain of (a,d)vertex-antimagic total labelings for

paths is studied as follows .

In P4 the minimum weight is obtained using formula (5) in chapter 3.3.1,

, where is the smallest degree in a graph. For P4 =1. Therefore,

40

. Thus

And the maximum is obtained using formula (5) in chapter 3.3.1,

“Similarly if φ is the largest degree, then the maximum vertex weight is no more than the

sum of φ + 1 largest labels”

For P4 φ=2. Therefore, sum of largest φ + 1 = 2 + 1 = 3 labels gives us maximum vertex

weight. Sum of largest 3 labels for P4 is 7 + 6 + 5 = 18. The smallest and largest weight that

can be obtained is shown in Figure 4.18.

To decide on the range of the increment, d, assume that the path contains the smallest

weight 3. If the increment d is 5, the set of weight is W = {3, 8, 13, 18}. All of the weights

are in the possible range of . In path P4 the increment d = 6 with a = 3 creates

a set of weights W = {3, 9, 15, 21}. A weight “21” is not in the range .

Therefore it is unnecessary to search the labelings with increment .

Boundaries of the increment constant, d, is calculated from (7) in chapter 3.3.1

For P4, M = v+e = 4+3 = 7, largest degree, = 2, smallest degree, = 1 and v = 4

In addition to the limitations on the range of increment, a range on the smallest or starting

weight a should be set. Since a is the smallest weight, the lower bound is the minimum

possible weight on a path P4 that is 3. The upper bound of a is determined by

2 3 5 4

1 6 710 18 113

Figure 4.18: Maximum and minimum weights labelled on P4

41

In the path P4 the lower bound of a is 3 and the upper bound is 15. The lower bound of the

increment d is 1 and the upper bound is 5. In the table below the range of a,d are given for

paths.

Number

of e

Number

of v

a d Smallest

weight

Largest weight

3 4 3 18

4 5 3 24

5 6 3 30

6 7 3 36

7 8 3 42

Table 4.9: Search range of a and d, smallest and largest possible weights for paths

Table 4.9 shows the search domain of a and d. The range of a and d is used to restrict the

computer program when searching through labels to avoid unnecessary computations. In

the figure below some (a,d)vertex-antimagic total labeling of paths are shown.

42

In Figure 4.19 few (a,d)vertex-antimagic total labelings for P4 and P5 are shown.

All possible (a,d)-vertex-antimagic labelings of P4, P5, P6 and P7 are listed in Appendix B.

4.4 Summary

The number of tries that needs to be executed is calculated for each graph. For cycles, paths

and trees the number of elements in the graph differs, therefore the number of permutations

is also different. As the number of permutation increases the number of tries also increases

which results in an increase in computation time. In (a,d)vertex antimagic total labeling the

range of search domain is calculated and the source program’s algorithm is produced

according to the search domain.

1 4 6 3

2 7 513 18 83

The only labeling of P4 where a=3 and d=5. This is called a (3,5)-vertex-antimagic total labeling

7 2 1 4

3 6 511 12 910

Labeling of P4 where a =9 and d=1.Out of all possible labelings, this is the one where a gets its highest value.

A Labeling of P5 where a=3 and d=5. This is called a (3,5)-vertex-antimagic total labeling2 3 5 8

1 4 98 18 233

7

6 13

7 6 3 8

2 5 913 17 219

1

4 5A Labeling of P5 where a=5 and d=4. This is called a (5,4)-vertex-antimagic total labeling

Figure 4.19: Examples of (a,d)vertex-antimagic total labelings for P4 and P5

43

CHAPTER 5: RESULTS

5.1 Overview

After the possible labelings are found, some open problems are addressed looking at the

results. The results found on vertex magic total labelings, (a,d)vertex antimagic total

labelings and the comparison between two processors on computation times are discussed

in this chapter.

5.2 Results on Vertex-magic Total Labelings

In this thesis the number of vertex-magic total labelings and the magic number(k) that is

found is given in table below.

Graph k (magic number) Number of

solutions

Graph k (magic

number)

Number of

solutions

C4 k=12 1 C5 k=14 1

k=13 2 k=16 2

k=14 2 k=17 2

k=15 1 k=19 1

All k (total) 6 All k (total) 6

C6 k=17 2 C7 k=19 1

k=18 1 k=20 5

k=19 5 k=21 9

k=20 5 k=22 17

k=21 1 k=23 17

k=22 2 k=24 9

All k (total) 16 k=25 5

44

k=26 1

All k (total) 64

P4 k=9 1 P5 k=11 1

k=10 2 k=13 2

k=11 1 k=14 1

All k (total) 4 All k (total) 4

P6 k=14 2 P7 k=16 1

k=15 1 k=17 5

k=16 3 k=18 7

k=17 1 k=19 11

All k (total) 7 k=20 8

k=21 3

All k (total) 35

Binary Tree

v=7,e=6

- 0

Binary Tree

v=8,e=7

k=19 1

k=20 7

k=21 4

All k (total) 12

Table 5.10: Summary of vertex-magic total labeling solutions for cycles, paths and trees

From the summary in Table 5.10 open problem of Cunningham in [12] can be answered;

“For a given cycle graph, is there a vertex-magic labeling associated with every magic

number within the bounds?”

From [12] the lower bound and upper bound of k in a cycle graph with 5 vertices is;

, for C5 v=5. Therefore

45

Looking at Table 5.10 we can see that C5 does not have a vertex-magic

total labeling with k=15 and k=18 even though they are in the boundaries.

5.3 Results on (a,d)Vertex-AntiMagic Total Labelings

The number of computer found (a,d)vertex-antimagic total labelings are in Appendix B.

The open problem stated by Baca, Bertault, McDougall, Miller, Simanjuntak and Slamin

[2]: “For the paths Pn and the cycles Cn, determine if there is a vertex-antimagic total

labeling for every feasible pair (a,d)” is addressed.

In section 3.2.1 the restrictions of a and d are stated as;

Also the formula (6) in section 3.3.1 brings further restrictions to a and d in some graphs.

For C4, Let a = 6. As known M=8, v=4, e=4, =2, Therefore

(Restriction 1)

Further restrictions on d are,

Since a = 6, then (Restriction 2)

Combining Restriction 1 and 2,

Therefore in C4, feasible pairs of (a,d) for a = 6 are (6,4) and (6,5). There are no

(6,4)vertex-antimagic total labelings for C4 even though it is feasible. Therefore there are

no vertex-antimagic total labelings for every feasible pair of (a,d).

5.4 Comparison of Computation Times for Two Different Processors

46

Systematic searches are completed in two different Central Processing Units to understand

the advantages of faster CPUs. A computer with an AMD Athlon 850Mhz CPU and a

computer with a Pentium 4 2300Mhz CPU is used. The programs were also executed in a

dual-core AMD x 2 processor but it is observed that only one processor executed the

program since the compiler didn’t support a dual-core processor. For this reason the dual-

core processor was not compared to the others.

The expected increase in speed of calculations is for 2300Mhz

CPU. In the calculations of vertex-magic total labelings calculations an average speedup of

3.2 times is observed which is more than the expected speedup. In the calculations of

(a,d)vertex-antimagic total labelings a speedup of 2.68 times is observed which is very

close to the expected speedup. Therefore it is obvious that the speed of a CPU directly

affects the speed of execution times of programs. Therefore it is inescapable to use latest

technology in such researches with large search domains such as DNA representations and

many other research areas.

The computation times for the cycle and path graphs are represented below.

VERTEX-MAGIC VERTEX-ANTIMAGIC

Graph Type and number of vertices Calculations

AMD 850MHz

(seconds)

Pentium 2300Mhz (seconds)

AMD 850MHz

(seconds)

Pentium 2300Mhz(seconds)

P4 5040 0.010 0.000 0.01 0.000C4 40320 0.020 0.000 0.08 0.031P5 362880 0.260 0.062 0.821 0.312C5 3628800 1.573 0.578 8.472 2.969P6 39916800 20.399 7.360 87.015 34.047C6 479001600 281.985 88.750 1159.93 406.344P7 6227020800 4668.190 1484.280 22187.200 8501.030

Table 5.11 Computation times of programs in two processors

The table below shows the average speedup of 2300Mhz processor compared to the

850Mhz processor.

GraphVMTL-average

speedupVATL-average

speedupP,v=4 Can’t be determined Can’t be determined

47

C,v=4 Can’t be determined 2.580645161P,v=5 4.193548387 2.631410256C,v=5 2.721453287 2.853486022P,v=6 2.771603261 2.555731783C,v=6 3.177295775 2.854551808P,v=7 3.14508718 2.60994256

Average 3.201797578 2.680961265Table 5.12: Speedup of 2300Mhz processor over 850Mhz processor

The data in Table 5.12 is summarized in a line chart as shown in Figure 5.20. A trend line

is added to the graph to show the constant speedup gain in vertex-antimagic total

calculations.

Average Speedup of 2300Mhz Processor over 850Mhz Processor

y = 0.0148x + 2.6291

00.5

11.5

22.5

33.5

44.5

C,v=4 P,v=5 C,v=5 P,v=6 C,v=6 P,v=7Graphs

Avera

ge Sp

eedo

ver

Calculations of Vertex-Magic Total Labelings

Calculations of Vertex-Antimagic TotalLabelingsLinear (Calculations ofVertex-Antimagic TotalLabelings)

Figure 5.20: Speedup of 2300Mhz processor over the 850Mhz processor for VMTL and VATL searches

As seen from Figure 5.20 the trend line is almost parallel to the X-axis which shows that

there is a constant average speedup of 2.68 in calculations of vertex-antimagic total

labelings. Therefore the speed of a processor is directly proportional to the calculation

speed of the computer program.

5.5 Summary

48

Few open problems that were the focus of this thesis is addressed and some are still left

unaddressed. The possible solutions for vertex magic total labelings and (a,d)vertex

antimagic total labelings are to be used in further research in this area.

CONCLUSION

Computer-assisted proofs and computer-assistance in many fields is a requirement

nowadays. Mathematicians have been constructing many useful theorems with pure

reasoning in many areas. Computers are fast, durable and very accurate devices that should

be used in experimental mathematics and any other area. A solution to a given problem can

be found by trying all possibilities to the problem. This thesis focuses on magic labelings in

the area of graph theory. In this thesis, a brute force algorithm with a pre-defined search

domain is used to find all possible vertex-magic or (a,d)-vertex antimagic total labeling of

cycles, paths and trees. In our algorithm, all possible labelings in the search domain are

tried and checked to see if it is a vertex-magic or a (a,d)vertex-antimagic labeling.

These results are then used to answer some open problems in this field of area such as “Is

there a vertex-magic total labeling of a cycle with all feasible values of k.” This question is

answered by looking at the vertex magic total labelings found for all cycles. Looking at

Table 5.10 realize that for C5 the minimum magic number is 14 and the maximum magic

number is 19. Therefore look if there is a vertex magic total labeling for k=15,16 and 17,

and realize that k=15 and k=17 has no possible VMTL. For C3,C4,C6,C7 there is a VMTL

with all feasible values of k. Therefore further research has to be done in order to check if

C5 is the only case which does not have all feasible VMTL.

Also the problem “Is there a (a,d)-vertex antimagic total labeling of a cycle with all feasible

pairs of (a,d).” is addressed. Looking at the results in Appendix B all possible pairs of (a,d)

VATL, this problem is answered. In C4, feasible pairs of (a,d) includes (6,4) and (6,5).

There are no (6,4)vertex-antimagic total labelings for C4 even though it is feasible.

Therefore there are no vertex-antimagic total labelings for every feasible pair of (a,d).

49

Tezer’s proof of binary trees is also confirmed with a computer search of all possibilities

and a binary tree with an added leaf is studied on.

The lack of computer power limited the research to be done. Two different processors with

different speeds (Mhz) are compared to see the rate of decrease in computation times.

An open problem from [12] that is not answered yet is raised again in this thesis with all

known vertex-magic total labeling results; Can an algorithm be created for finding all

possible vertex-magic total labelings for a given cycle graph?

The same problem is stated for (a,d)-vertex antimagic total labelings as an open problem in

this thesis; Can an algorithm be created for finding all possible feasible pairs of (a,d) for a

given cycle graph?

Future works include finding another algorithm that will solve the problem faster and more

efficient. I plan to continue this research with the help of a faster algorithm and faster

CPUs. I also plan to research how to run multiple computers in parallel to add more CPU

power on a specific problem. I plan to study this theoretical problem in a mathematical way

and not only with computer programming to present a permanent algorithm.

50

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[47] Sun, G. C.; Lee, S. M. (1994). Construction of magic graphs. Congress. Numerantium, 103: 935-939.

[48] Swaminathan, V.; Jeyanthi, P. (2003). Super vertex-magic labeling. Indian J. Pure Appl. Math., 34: 935-939.

[49] Tezer, M.; Cahit, I. (2000). A Note on Vertex Magic Total Labeling of a Class of Trees. XIII National Mathematic Symposium, Boğaziçi Bilgi Sabancı University İstanbul.

[50] Tezer, M.; Cahit, I. (2005). A note on (a,d)-Vertex Antimagic Total Labeling of Path and Cycles. Utilitas Mathematica, 68:217-221.

[51] Trenkler, M. (2000). Numbers of vertices and edges of magic graphs. Ars. Combin., 55: 93-96.

[52] Wallis, W.D. (2001). Magic Graphs. New York: Birkhauser Boston.

[53] Wood, D. R. (2002). On vertex-magic and edge-magic total injections of graphs. Australasian Journal of Combin., 26: 49-63.

54

APPENDIX A (VERTEX-MAGIC TOTAL LABELINGS):

The solutions are represented as sets only. The edges and vertices can be labeled on the

graph in the represented order to obtain a vertex-magic graph.

A1: VERTEX-MAGIC TOTAL LABELINGS FOR CYCLES

V,E = 3Vertex-Magic Labelings for Cycles with e,v = 3

Solution Number E = {e1,e2,e3} V = {v1,v2,v3} Magic number k

1 2.1.3 4.6.5 k=9 2 3.1.5 2.6.4 k=10 3 4.2.6 1.5.3 k=11 4 5.4.6 1.3.2 k=12

V,E = 4Vertex-Magic Labelings for Cycles with e,v = 4

Solution Number E = {e1,e2,e3,e4} V = {v1,v2,v3,v4} Magic number k

1 1.3.2.6 5.8.7.4 k=12 2 1.5.2.8 4.7.6.3 k=13 3 1.4.6.5 7.8.3.2 k=13 4 1.7.4.8 5.6.3.2 k=14 5 3.4.8.5 6.7.2.1 k=14 6 3.7.6.8 4.5.2.1 k=15

V,E = 5Vertex-Magic Labelings for Cycles with e,v = 5

Solution Number E = {e1,e2,e3,e4,e5} V = {v1,v2,v3,v4,v5} Magic number k

v1 v2

v3v4

e4

e1

e2

e3

55

1 1.3.5.2.4 9.10.6.7.8 k=14 2 1.7.3.4.10 5.8.6.9.2 k=16 3 1.5.9.3.7 8.10.2.4.6 k=16 4 1.7.8.4.10 6.9.2.5.3 k=17 5 2.6.10.4.8 7.9.1.3.5 k=17 6 6.8.10.7.9 4.5.1.2.3 k=19

V,E = 6Vertex-Magic Labelings for Cycles with e,v = 6

Solution Number E = {e1,e2,e3,e4,e5,e6} V = {v1,v2,v3,v4,v5,v6} Magic number k

1 1.5.4.3.2.9 7.11.8.10.12.6 k = 17 2 1.5.2.3.6.7 9.11.10.12.8.4 k = 17 3 1.8.4.2.5.10 7.9.6.12.11.3 k = 18 4 1.8.9.4.3.11 7.10.2.6.12.5 k = 19 5 1.7.3.12.5.8 10.11.9.4.2.6 k = 19 6 1.6.11.3.7.8 10.12.2.5.9.4 k = 19 7 1.8.5.12.6.10 9.11.7.3.2.4 k = 20 8 2.7.11.3.4.9 8.10.1.5.12.6 k = 19 9 2.10.9.4.5.12 6.8.1.7.11.3 k = 20 10 2.6.11.4.9.10 8.12.3.5.7.1 k = 20 11 2.7.12.5.6.10 8.11.1.3.9.4 k = 20 12 2.6.10.9.8.7 11.12.4.1.3.5 k = 20 13 3.4.5.6.11.7 9.12.10.8.2.1 k = 19 14 3.8.11.9.5.12 6.10.2.1.7.4 k = 21 15 4.11.10.9.8.12 6.7.1.3.5.2 k = 22 16 6.7.10.11.8.12 4.9.5.1.3.2 k = 22

V,E = 7Vertex-Magic Labelings for Cycles with e,v = 7

Solution Number E={e1,e2,e3,e4,e5,e6,e7} V={v1,v2,v3,v4,v5,v6,v7} Magic number k

1 1.4.2.5.6.3.7 11.14.13.12.8.10.9 k = 192 1.5.4.3.8.2.12 7.14.11.13.9.10.6 k = 203 1.9.4.2.5.3.11 8.10.7.14.13.12.6 k = 204 1.5.2.6.10.3.8 11.14.13.12.4.7.9 k = 205 1.8.2.9.5.3.14 6.12.11.10.7.13.4 k = 216 1.7.6.3.14.2.9 11.13.8.12.4.5.10 k = 217 1.11.2.12.6.3.14 7.10.9.8.4.13.5 k = 228 1.5.4.7.10.2.6 13.14.11.9.3.8.12 k = 209 1.5.2.6.4.9.8 11.14.13.12.10.7.3 k = 20

10 1.11.2.12.4.5.14 7.10.9.8.6.13.3 k = 2211 1.11.4.13.6.2.12 9.10.7.5.3.14.8 k = 22

56

12 1.7.6.12.5.2.9 11.13.8.3.4.14.10 k = 2113 1.10.5.13.6.2.12 9.11.7.4.3.14.8 k = 2214 1.7.2.5.10.8.9 11.13.12.14.6.3.4 k = 2115 1.9.5.14.2.7.11 10.12.8.3.6.13.4 k = 2216 1.8.10.5.14.2.9 12.13.4.7.3.6.11 k = 2217 1.7.3.8.4.5.14 6.13.11.10.9.12.2 k = 2118 1.6.13.3.7.4.8 12.14.2.5.11.10.9 k = 2119 1.12.9.3.13.4.14 8.10.2.11.7.6.5 k = 2320 1.10.11.3.13.4.14 8.12.2.9.7.6.5 k = 2321 1.7.5.13.3.8.12 9.14.10.4.6.11.2 k = 2222 1.7.3.9.11.5.13 8.14.12.10.2.6.4 k = 2223 1.6.5.4.8.11.7 13.14.10.12.9.2.3 k = 2124 1.8.5.14.6.4.11 10.13.9.3.2.12.7 k = 2225 1.9.6.14.5.4.10 11.12.7.2.3.13.8 k = 2226 1.7.12.4.5.9.11 10.14.3.6.13.8.2 k = 2227 1.10.6.14.4.8.13 9.12.7.3.5.11.2 k = 2328 1.9.11.4.14.7.10 12.13.3.8.5.2.6 k = 2329 1.10.5.7.14.6.13 9.12.8.11.2.3.4 k = 2330 1.9.6.14.5.11.10 12.13.8.3.4.7.2 k = 2331 1.8.12.5.11.10.9 13.14.3.6.7.2.4 k = 2332 1.8.6.13.7.11.10 12.14.9.4.3.5.2 k = 2333 1.12.10.6.13.7.14 9.11.2.8.5.4.3 k = 2434 1.12.9.8.14.6.13 10.11.3.7.2.4.5 k = 2435 1.10.11.7.12.8.14 9.13.3.6.5.4.2 k = 2436 2.5.4.8.3.7.13 6.14.12.9.10.11.1 k = 2137 2.6.3.4.7.9.11 8.13.12.14.10.5.1 k = 2138 2.9.13.3.14.4.11 10.12.1.7.6.5.8 k = 2339 2.11.10.5.3.6.12 8.9.1.7.14.13.4 k = 2240 2.8.13.3.5.7.11 9.12.1.6.14.10.4 k = 2241 2.7.11.5.3.9.12 8.13.4.6.14.10.1 k = 2242 2.9.13.3.14.5.10 11.12.1.7.6.4.8 k = 2343 2.10.8.14.3.7.12 9.11.5.1.6.13.4 k = 2344 2.8.4.5.14.7.9 11.12.10.13.3.1.6 k = 2245 2.8.4.10.12.6.14 7.13.11.9.1.5.3 k = 2346 2.7.13.4.8.10.12 9.14.3.6.11.5.1 k = 2347 2.9.14.7.11.8.12 10.13.1.3.6.5.4 k = 2448 2.8.10.11.12.7.13 9.14.6.3.1.5.4 k = 2449 3.7.14.6.5.4.10 9.12.1.2.11.13.8 k = 2250 3.5.4.10.11.9.7 12.14.13.8.1.2.6 k = 2251 3.9.12.10.5.4.13 7.11.2.1.8.14.6 k = 2352 3.8.14.4.6.10.11 9.12.1.5.13.7.2 k = 2353 3.6.12.10.4.8.13 7.14.5.1.9.11.2 k = 2354 3.9.8.14.6.13.10 11.12.7.2.4.5.1 k = 2455 3.13.7.12.11.10.14 8.9.5.6.2.4.1 k = 2556 4.5.11.10.12.8.6 13.14.7.2.1.3.9 k = 2357 4.6.8.11.12.9.13 7.14.10.5.1.3.2 k = 2458 4.12.10.13.11.6.14 7.9.3.2.1.8.5 k = 2559 4.7.11.10.9.14.8 12.13.6.3.5.1.2 k = 2460 5.7.6.14.8.13.10 9.12.11.4.2.3.1 k = 24

57

61 5.9.13.10.14.7.12 8.11.3.2.1.4.6 k = 2562 5.8.10.11.13.9.14 6.12.7.4.1.3.2 k = 2563 6.7.10.11.13.9.14 5.12.8.4.1.3.2 k = 2564 8.11.9.12.13.10.14 4.7.6.5.1.3.2 k = 26

A2: VERTEX-MAGIC LABELINGS FOR PATHS

V=4, E=3Vertex-Magic Labelings for Paths with v=4,e=3.

Solution Number E={e1,e2,e3} V={v1,v2,v3,v4} Magic number k

1 2.1.5 7.6.3.4 k = 9 2 4.1.7 6.5.2.3 k = 10 3 3.5.4 7.2.1.6 k = 10 4 6.3.7 5.2.1.4 k= 11

V=5, E=4Vertex-Magic Labelings for Paths with v=5,e=4.

Solution Number E={e1,e2,e3,e4} V={v1,v2,v3,v4,v5} Magic number k

1 2.4.1.3 9.5.6.7.8 k = 11 2 6.2.3.9 7.5.8.1.4 k = 13 3 4.8.2.6 9.1.3.5.7 k = 13 4 6.7.3.9 8.1.4.2.5 k = 14

V=6, E=5Vertex-Magic Labelings for Paths with v=6,e=5.

Solution Number E={e1,e2,e3,e4,e5} V={v1,v2,v3,v4,v5,v6} Magic number k

1 4.3.2.1.8 10.7.9.11.5.6 k = 14 2 4.1.2.5.6 10.9.11.7.3.8 k = 14 3 7.3.1.4.9 8.5.11.10.2.6 k = 15 4 7.8.3.2.10 9.1.5.11.4.6 k = 16 5 6.2.11.4.7 10.8.3.1.5.9 k = 16 6 5.10.2.6.7 11.1.4.8.3.9 k = 16 7 7.4.11.5.9 10.6.2.1.3.8 k = 17

V1 V2 V3 V4

E1 E2 E3

58

V=7, E=6Vertex-Magic Labelings for Paths with v=7,e=6.

Solution Number E={e1,e2,e3,e4,e5,e6} V={v1,v2,v3,v4,v5,v6,v7} Magic number k

1 3.1.4.5.2.6 13.12.11.7.9.8.10 k = 16 2 4.3.2.7.1.11 13.10.12.8.9.5.6 k = 17 3 8.3.1.4.2.10 9.6.13.12.11.5.7 k = 17 4 4.1.5.9.2.7 13.12.11.3.6.8.10 k = 17 5 7.1.8.4.2.13 11.10.9.6.12.3.5 k = 18 6 6.5.2.13.1.8 12.7.11.3.4.9.10 k = 18 7 10.1.11.5.2.13 9.8.7.3.12.4.6 k = 19 8 4.3.6.9.1.5 13.10.8.2.7.11.12 k = 17 9 4.1.5.3.8.7 13.12.11.9.6.2.10 k = 17 10 10.1.11.3.4.13 9.8.7.5.12.2.6 k = 19 11 10.3.12.5.1.11 9.6.4.2.13.7.8 k = 19 12 6.5.11.4.1.8 12.7.2.3.13.9.10 k = 18 13 9.4.12.5.1.11 10.6.3.2.13.7.8 k = 19 14 6.1.4.9.7.8 12.11.13.5.2.3.10 k = 18 15 8.4.13.1.6.10 11.7.2.5.12.3.9 k = 19 16 7.9.4.13.1.8 12.3.6.2.5.10.11 k = 19 17 6.2.7.3.4.13 12.10.9.8.11.1.5 k = 18 18 5.12.2.6.3.7 13.1.4.10.9.8.11 k = 18 19 11.8.2.12.3.13 9.1.10.6.5.4.7 k = 20 20 9.10.2.12.3.13 11.1.8.6.5.4.7 k = 20 21 6.4.12.2.7.11 13.9.3.5.10.1.8 k = 19 22 6.2.8.10.4.12 13.11.9.1.5.3.7 k = 19 23 5.4.3.7.10.6 13.9.11.8.1.2.12 k = 18 24 7.4.13.5.3.10 12.8.2.1.11.6.9 k = 19 25 8.5.13.4.3.9 11.6.1.2.12.7.10 k = 19 26 6.11.3.4.8.10 13.2.5.12.7.1.9 k = 19 27 9.5.13.3.7.12 11.6.2.4.10.1.8 k = 20 28 8.10.3.13.6.9 12.2.7.4.1.5.11 k = 20 29 9.4.6.13.5.12 11.7.10.1.2.3.8 k = 20 30 8.5.13.4.10.9 12.7.2.3.6.1.11 k = 20 31 7.11.4.10.9.8 13.2.5.6.1.3.12 k = 20 32 7.5.12.6.10.9 13.8.3.2.4.1.11 k = 20 33 11.9.5.12.6.13 10.1.7.4.3.2.8 k = 21 34 11.8.7.13.5.12 10.2.6.1.3.4.9 k = 21 35 9.10.6.11.7.13 12.2.5.4.3.1.8 k = 21

59

A3: VERTEX-MAGIC LABELINGS FOR BINARY TREES

V=7, E=6Vertex-Magic Labelings for Tree with v=7,e=6.

Solution Number E={e1,e2,e3,e4,e5,e6} V={v1,v2,v3,v4,v5,v6,v7} Magic number k

No solutions found

V=8, E=7Vertex-Magic Labelings for Tree with v=8,e=7.

Solution Number E={e1,e2,e3,e4,e5,e6,e7} V={v1,v2,v3,v4,v5,v6,v7,v8} Magic number k

1 1.3.2.6.5.7.8 15.10.9.13.4.14.12.11 k = 19 2 2.3.1.12.6.7.9 15.5.10.8.4.14.13.11 k = 20 3 4.2.1.12.5.7.9 14.3.10.8.6.15.13.11 k = 20 4 10.2.1.5.6.9.7 8.4.12.15.3.14.11.13 k = 20

V1

V2

V3 V4

V5

V6 V7

E1 E2

E3 E4 E5 E6

V1

V2

V3 V4

V5

V6 V7

E1 E2

E3 E4 E5 E6

V8

E7

60

5 5.1.2.11.7.9.13 15.3.6.10.4.14.12.8 k = 21 6 3.4.1.6.5.9.12 13.10.7.14.2.15.11.8 k = 20 7 1.4.3.11.6.8.7 15.5.10.9.2.14.12.13 k = 20 8 10.1.3.5.7.8.6 9.2.11.15.4.13.12.14 k = 20 9 2.4.3.5.7.8.11 14.10.6.15.1.13.12.9 k = 20 10 2.4.3.10.7.9.13 15.6.5.11.1.14.12.8 k = 21 11 4.3.7.8.6.11.9 14.2.5.13.1.15.10.12 k = 21 12 5.3.8.6.7.10.9 13.2.4.15.1.14.11.12 k = 21

APPENDIX B ((a,d)-VERTEX-ANTIMAGIC TOTAL LABELINGS):

The solutions are represented in sets only. The edges and vertices can be labeled on the

graph in the represented order to obtain a (a,d)vertex-antimagicmagic graph.

61

B1: (a,d)-VERTEX-ANTIMAGIC TOTAL LABELINGS FOR CYCLES

V,E=4Vertex-AntiMagic Labelings for Cycle with v,e=4.

a d E={e1,e2,e3,e4} V={v1,v2,v3,v4}

6 5 1.3.8.6 4.2.5.77 4 2.3.7.4 1.6.5.88 3 1.2.8.3 7.5.4.68 3 1.2.4.7 3.5.8.68 3 2.3.4.5 1.6.7.89 2 1.2.6.3 5.8.7.49 2 1.2.4.5 3.8.7.69 3 1.2.8.7 4.6.5.39 3 1.4.8.5 3.7.6.29 3 3.4.6.5 1.8.2.7

10 1 1.2.3.4 5.8.7.610 2 1.2.8.5 6.7.4.310 2 1.6.2.7 4.3.8.510 2 1.3.8.4 7.6.5.210 2 1.3.5.7 2.8.6.410 2 1.5.4.6 3.8.7.210 2 2.5.3.6 8.7.4.110 3 1.6.8.7 5.3.2.410 3 4.5.6.7 2.1.8.311 1 1.2.6.5 7.8.4.311 1 1.3.6.4 8.7.5.211 2 1.4.8.7 5.6.3.211 2 1.5.6.8 2.7.4.311 2 2.7.3.8 1.4.5.611 2 2.4.6.8 1.7.5.311 2 2.5.7.6 3.8.1.411 2 3.5.4.8 6.7.2.111 2 3.6.4.7 1.8.5.212 1 1.4.7.6 5.8.3.212 1 2.3.8.5 6.7.4.112 1 2.3.7.6 4.8.5.112 1 2.4.5.7 3.8.6.1

v1 v2

v3v4

e4

e1

e2

e3

62

12 2 3.6.8.7 2.5.4.112 2 4.5.7.8 2.3.6.113 1 2.5.7.8 4.6.3.113 1 3.4.8.7 5.6.2.113 1 3.5.8.6 4.7.1.214 1 5.6.7.8 1.4.3.2

V,E=5Vertex-AntiMagic Labelings for Cycle with v,e=5.

a d E={e1,e2,e3,e4,e5} V={v1,v2,v3,v4,v5}

6 5 1.2.5.10.7 8.3.4.6.96 5 1.2.9.7.6 4.3.5.10.87 5 2.4.5.10.9 6.1.3.7.88 4 1.3.2.10.9 6.4.7.8.58 4 1.3.6.10.5 2.8.7.4.99 3 1.2.4.3.10 7.6.9.5.89 3 1.2.4.7.6 8.9.3.10.59 3 1.3.4.7.5 9.8.2.10.69 3 2.3.4.5.6 1.7.8.9.109 4 1.2.10.8.9 3.6.5.7.49 4 1.5.8.6.10 2.3.4.7.910 3 1.3.4.10.7 5.6.9.8.210 3 1.4.5.9.6 3.8.10.2.710 3 2.3.6.4.10 1.5.7.9.810 4 3.5.8.10.9 6.2.1.4.711 2 2.3.5.4.6 7.8.9.10.111 3 1.4.7.10.8 2.9.6.3.511 3 2.3.10.7.8 1.9.4.6.512 1 1.2.3.5.4 7.10.9.8.612 2 1.5.8.2.9 4.6.3.10.712 2 1.3.8.4.9 2.10.5.6.712 2 1.3.5.9.7 4.10.8.6.212 2 1.3.6.8.7 4.10.9.2.512 2 3.4.6.5.7 8.9.10.1.212 3 1.7.8.10.9 5.4.3.6.212 3 2.6.8.10.9 1.7.4.3.512 3 5.6.8.9.7 3.1.10.4.213 1 1.2.5.3.9 7.10.8.6.413 1 1.2.6.3.8 5.10.9.7.413 1 1.2.8.5.4 9.10.7.3.613 2 1.4.6.10.9 5.8.7.3.213 2 2.6.9.3.10 1.7.4.5.813 2 2.4.6.10.8 3.9.7.5.113 2 4.5.7.6.8 1.10.9.2.314 1 1.4.8.2.10 5.9.3.7.614 1 1.4.5.8.7 6.10.9.3.2

63

14 1 2.3.7.4.9 5.10.8.6.114 1 2.3.6.9.5 7.10.8.1.414 1 2.3.5.8.7 6.9.10.4.114 1 2.4.5.8.6 7.10.9.1.315 1 1.7.9.3.10 4.8.2.5.615 1 1.5.9.7.8 6.10.3.2.415 1 2.4.7.9.8 5.10.6.3.115 1 2.5.6.10.7 8.9.4.3.115 1 2.5.8.6.9 7.10.3.1.415 1 3.4.5.10.8 7.9.6.2.115 1 3.4.6.10.7 5.9.8.1.215 1 3.4.6.9.8 7.10.5.1.215 1 3.5.9.7.6 10.8.4.1.216 1 3.6.7.10.9 4.8.5.2.116 1 4.5.7.10.9 3.8.6.2.116 1 4.6.7.10.8 5.9.3.1.217 1 6.7.8.10.9 2.5.4.3.1

V,E=6Vertex-AntiMagic Labelings for Cycle with v,e=6.

a d E={e1,e2,e3,e4,e5,e6} V={v1,v2,v3,v4,v5,v6}

6 5 1.2.4.12.8.6 9.3.5.10.11.76 5 1.2.5.6.11.8 7.3.4.10.9.127 5 1.2.3.11.12.10 6.4.7.8.9.57 5 1.2.5.11.12.8 3.4.10.6.9.77 5 1.2.12.9.10.5 6.4.3.11.8.77 5 1.4.3.8.12.11 10.2.5.6.7.97 5 2.4.5.7.12.9 6.1.3.10.8.118 4 2.3.4.5.10.6 8.7.1.11.9.128 5 1.5.7.12.11.9 3.2.6.4.10.88 5 2.5.7.12.11.8 3.1.6.4.10.99 4 1.2.7.12.5.9 3.6.8.10.4.119 4 1.5.2.7.10.11 9.3.6.8.12.49 4 1.2.11.8.5.9 3.6.4.10.12.79 4 1.3.6.4.12.10 2.5.8.11.9.79 4 1.3.4.7.12.9 11.5.10.2.6.89 4 1.4.11.6.9.5 3.8.2.12.10.79 4 1.4.7.8.11.5 3.12.2.6.10.99 4 1.4.7.10.9.5 3.8.6.12.2.119 4 2.3.5.9.6.11 4.8.1.7.10.129 4 2.4.7.5.8.10 1.3.6.9.12.119 4 3.4.7.5.9.8 6.2.10.1.11.1210 3 1.2.4.6.5.9 3.7.10.12.8.1110 3 2.3.4.5.6.7 1.8.9.10.11.1210 3 2.3.4.5.7.6 11.8.9.1.10.1210 4 1.2.11.6.12.10 3.7.5.9.4.8

64

10 4 1.2.9.10.8.12 5.7.3.11.4.610 4 1.3.10.12.11.5 8.6.9.4.7.210 4 1.3.7.9.12.10 11.6.4.2.5.810 4 1.3.8.10.11.9 4.6.7.12.5.210 4 1.4.12.6.9.10 3.5.2.8.7.1110 4 1.4.8.12.7.10 3.5.6.2.11.910 4 1.5.7.10.8.11 2.4.6.9.12.310 4 2.3.6.11.8.12 4.9.1.5.7.1010 4 2.3.10.9.6.12 4.5.1.11.7.810 4 2.3.10.7.12.8 4.5.9.1.11.610 4 2.5.6.12.10.7 1.11.3.4.8.910 4 2.5.6.9.12.8 4.3.7.11.1.1010 4 2.5.6.11.10.8 4.3.7.9.1.1210 4 3.5.8.6.9.11 4.2.1.12.7.1010 4 4.5.6.9.10.8 2.1.7.11.3.1211 3 1.2.3.11.10.6 7.8.12.9.5.411 3 1.2.3.7.11.9 4.8.12.10.5.611 3 1.2.9.4.12.5 11.8.3.7.10.611 3 1.2.5.6.12.7 3.11.10.9.8.411 3 1.2.5.9.6.10 12.8.7.3.11.411 3 1.2.9.7.8.6 4.11.12.10.5.311 3 1.3.5.4.8.12 7.10.9.2.11.611 3 2.3.6.4.10.8 1.9.11.7.12.511 3 3.4.5.7.6.8 9.10.2.11.1.1211 4 2.4.9.11.12.10 3.5.6.7.8.111 4 2.6.8.11.9.12 1.3.5.4.7.1011 4 3.6.9.12.7.11 1.2.4.10.8.511 4 4.5.7.11.12.9 10.2.3.1.8.612 3 1.4.2.11.9.12 5.10.6.8.7.312 3 1.2.6.7.12.11 3.9.10.8.5.412 3 1.2.6.10.8.12 5.9.7.11.3.412 3 1.2.6.10.11.9 8.12.4.5.3.712 3 1.7.2.8.9.12 11.10.3.5.4.612 3 1.2.7.8.11.10 4.9.12.3.5.612 3 1.3.4.9.12.10 7.11.5.8.6.212 3 1.3.11.7.5.12 2.8.4.6.9.1012 3 1.3.5.10.12.8 9.11.4.6.2.712 3 1.3.5.10.9.11 6.8.7.12.2.412 3 1.3.6.7.12.10 4.8.9.11.2.512 3 1.7.3.9.8.11 6.4.5.12.10.212 3 1.4.5.11.6.12 2.7.9.8.10.312 3 1.4.8.5.12.9 11.7.6.2.10.312 3 1.4.5.10.8.11 3.7.9.12.6.212 3 1.4.6.7.12.9 2.10.11.5.8.312 3 2.3.4.11.12.7 9.10.5.6.1.812 3 2.3.10.8.4.12 1.7.5.6.9.1112 3 2.3.4.9.11.10 12.7.8.5.1.612 3 2.3.5.6.12.11 8.7.10.4.9.112 3 2.3.5.8.12.9 4.7.10.11.1.6

65

12 3 2.3.6.8.11.9 1.10.12.4.5.712 3 2.4.5.10.7.11 8.9.3.12.1.612 3 2.4.6.11.7.9 1.12.5.10.3.812 3 2.5.6.9.7.10 3.11.1.12.8.412 3 3.4.5.10.9.8 1.11.6.12.2.712 3 4.5.6.7.9.8 3.12.1.11.2.1013 2 1.2.3.6.11.7 9.12.8.10.4.513 2 1.2.6.8.10.3 9.12.11.7.5.413 2 1.2.3.8.9.7 11.10.12.4.6.513 2 1.2.4.12.6.5 11.10.9.7.3.813 2 1.2.5.4.11.7 9.10.12.6.8.313 2 1.2.4.10.8.5 7.12.11.9.3.613 2 1.2.4.10.6.7 5.12.11.9.3.813 2 1.2.6.8.4.9 3.12.11.7.5.1013 2 1.2.5.7.6.9 3.12.10.11.8.413 2 1.3.4.5.11.6 8.9.10.12.7.213 2 1.3.4.5.10.7 9.11.6.12.8.213 2 1.3.8.5.4.9 7.11.2.6.12.1013 2 1.3.4.7.9.6 10.11.12.2.5.813 2 1.3.6.5.7.8 4.11.10.12.9.213 2 2.3.4.7.5.9 6.8.12.10.11.113 2 2.3.7.4.8.6 5.12.9.10.11.113 3 1.9.2.12.10.11 7.3.5.8.6.413 3 1.3.8.10.12.11 7.9.5.4.6.213 3 1.6.12.5.11.10 2.9.4.8.3.713 3 1.5.8.9.12.10 11.7.3.2.4.613 3 1.6.11.7.12.8 4.9.2.10.3.513 3 1.6.8.10.11.9 3.12.2.4.7.513 3 2.3.7.12.11.10 4.8.9.6.5.113 3 2.4.10.6.12.11 3.7.8.9.1.513 3 2.4.9.8.10.12 11.7.3.5.1.613 3 2.5.7.10.12.9 8.6.4.11.3.113 3 2.6.7.9.11.10 4.5.12.3.8.113 3 3.4.10.11.5.12 1.6.8.7.9.213 3 3.4.7.11.8.12 10.6.5.1.9.213 3 3.5.7.9.11.10 6.8.1.12.2.413 3 5.6.7.9.8.10 1.2.12.3.11.414 2 1.3.7.12.2.11 4.10.8.5.6.914 2 1.2.8.10.3.12 5.11.6.4.7.914 2 1.3.11.2.10.9 4.12.8.7.6.514 2 1.4.6.11.2.12 3.9.8.5.7.1014 2 1.2.7.10.4.12 3.11.9.5.6.814 2 1.2.8.4.9.12 5.11.6.10.7.314 2 1.4.2.11.8.10 5.9.12.7.3.614 2 1.2.5.10.6.12 7.11.9.3.8.414 2 1.2.7.8.6.12 3.11.9.5.10.414 2 1.7.2.6.9.11 10.8.5.12.3.414 2 1.2.6.9.8.10 5.11.12.3.7.414 2 1.3.4.12.5.11 8.10.9.2.7.6

66

14 2 1.3.7.10.4.11 2.12.8.5.6.914 2 1.3.4.8.11.9 10.12.7.6.5.214 2 1.3.5.9.6.12 11.10.8.4.7.214 2 1.3.6.10.5.11 2.12.9.4.7.814 2 1.5.7.3.8.12 11.10.6.4.9.214 2 1.3.5.7.11.9 4.12.10.8.6.214 2 1.3.5.9.8.10 11.12.6.4.7.214 2 1.3.6.8.7.11 2.12.9.10.5.414 2 1.3.6.10.7.9 4.12.11.2.5.814 2 1.4.5.12.8.6 7.11.9.3.2.1014 2 1.4.6.10.7.8 5.11.12.2.3.914 2 1.5.7.6.8.9 4.12.10.11.2.314 2 2.3.4.10.5.12 6.9.11.8.1.714 2 2.3.4.12.9.6 10.11.7.8.1.514 2 2.3.6.10.11.4 12.9.7.8.1.514 2 2.3.4.8.7.12 6.9.11.10.1.514 2 2.4.7.9.3.11 1.12.5.6.8.1014 2 2.3.4.9.10.8 12.11.7.5.1.614 2 2.3.6.5.12.8 10.9.7.11.1.414 2 2.3.5.7.11.8 6.9.10.12.4.114 2 2.3.5.9.7.10 12.11.6.8.4.114 2 2.3.6.8.7.10 12.9.11.4.1.514 2 2.4.6.7.12.5 9.8.10.11.3.114 2 2.4.5.6.11.8 12.10.9.3.7.114 2 2.4.5.6.10.9 3.12.7.11.8.114 2 2.4.7.6.8.9 5.12.11.1.10.314 2 3.4.6.7.5.11 8.9.10.1.12.214 2 3.4.6.5.8.10 1.9.12.7.11.214 3 1.8.9.11.10.12 4.5.3.6.2.714 3 2.7.9.11.10.12 3.5.4.6.8.114 3 4.5.12.10.11.9 1.8.3.7.2.614 3 4.7.8.11.9.12 1.3.5.10.6.214 3 6.7.8.9.11.10 4.1.2.12.3.515 1 1.2.4.5.8.7 9.12.10.11.6.315 1 1.3.4.5.8.6 10.11.12.9.7.215 2 1.9.2.11.7.12 4.5.8.10.3.615 2 1.6.3.11.9.12 4.8.10.7.5.215 2 1.7.3.8.11.12 6.9.5.10.4.215 2 1.4.9.5.11.12 6.10.8.3.7.215 2 1.4.6.8.12.11 5.10.9.7.3.215 2 1.4.7.10.9.11 3.12.8.6.2.515 2 1.5.11.6.7.12 4.9.3.8.10.215 2 1.5.10.6.8.12 2.11.4.9.7.315 2 1.6.5.9.10.11 7.8.12.3.2.415 2 1.5.7.12.9.8 10.11.3.2.4.615 2 1.6.7.8.11.9 5.10.12.4.2.315 2 2.10.3.11.4.12 1.5.6.7.8.915 2 2.3.9.12.5.11 8.10.7.4.6.115 2 2.3.7.10.9.11 6.12.5.8.4.1

67

15 2 2.4.5.11.8.12 3.9.10.7.6.115 2 2.4.9.5.10.12 1.11.6.7.8.315 2 2.4.7.11.6.12 3.9.10.1.8.515 2 2.4.6.8.12.10 3.11.9.7.5.115 2 2.6.10.4.9.11 8.7.1.5.12.315 2 2.4.7.9.8.12 1.11.10.3.6.515 2 2.4.7.11.8.10 5.9.12.1.6.315 2 2.5.7.8.11.9 6.12.3.10.4.115 2 2.6.7.9.8.10 3.11.12.1.4.515 2 3.4.5.11.7.12 8.10.6.9.1.215 2 3.5.4.8.10.12 2.11.6.9.7.115 2 3.4.5.11.9.10 2.12.8.7.1.615 2 3.4.10.6.12.7 9.8.11.1.5.215 2 3.4.6.8.12.9 7.10.5.11.1.215 2 3.4.6.10.8.11 1.12.7.9.5.215 2 3.4.7.9.8.11 1.10.12.5.2.615 2 3.5.6.9.7.12 8.11.4.10.1.215 2 3.5.6.7.11.10 2.9.12.8.1.415 2 3.5.8.7.9.10 2.11.12.6.1.415 2 4.5.7.6.8.12 1.10.9.2.11.315 2 4.5.6.7.11.9 2.8.10.12.1.316 1 1.6.2.9.3.12 4.11.8.10.7.516 1 1.7.2.8.3.12 4.10.11.9.5.616 1 1.3.10.2.9.8 11.12.5.7.6.416 1 1.4.5.10.2.11 7.12.9.6.8.316 1 1.4.6.9.2.11 7.12.8.5.10.316 1 1.4.2.8.7.11 6.12.10.9.5.316 1 1.4.2.9.7.10 6.11.12.8.5.316 1 1.5.2.7.8.10 6.12.9.11.4.316 1 1.3.4.9.5.11 7.12.10.8.6.216 1 1.4.3.7.6.12 5.11.10.9.8.216 1 1.3.4.11.6.8 9.12.10.5.2.716 1 1.3.4.9.6.10 8.12.11.7.2.516 1 1.3.4.8.7.10 6.12.11.9.5.216 1 1.3.5.6.11.7 9.12.10.8.4.216 1 1.3.5.8.6.10 9.12.11.4.7.216 1 1.3.5.8.7.9 10.12.11.4.6.216 1 1.4.5.6.10.7 9.11.12.8.2.316 1 1.4.5.6.9.8 7.12.11.10.3.216 1 2.3.4.7.5.12 6.11.10.8.9.116 1 2.3.5.4.8.11 6.12.10.7.9.116 1 2.3.4.5.10.9 7.11.12.8.6.116 1 2.3.4.11.6.7 10.12.9.5.1.816 1 2.3.4.7.9.8 10.11.12.6.5.116 1 2.3.5.6.10.7 9.11.12.8.1.416 1 2.3.5.6.9.8 7.11.12.10.4.116 1 2.4.5.6.9.7 8.12.11.10.1.316 2 1.9.5.10.12.11 4.8.6.7.2.316 2 1.6.8.11.10.12 7.9.4.3.5.2

68

16 2 1.7.8.12.9.11 4.10.5.2.3.616 2 2.5.8.11.12.10 6.9.7.3.1.416 2 2.6.7.10.12.11 5.8.9.3.4.116 2 2.7.8.9.12.10 6.11.1.5.3.416 2 3.4.8.11.10.12 5.9.6.7.1.216 2 3.5.7.11.10.12 9.8.6.2.1.416 2 3.5.8.12.9.11 2.10.7.4.1.616 2 3.6.8.9.12.10 5.11.2.7.1.416 2 3.7.8.10.9.11 4.12.1.2.5.616 2 4.5.6.11.12.10 8.7.9.1.3.216 2 4.5.8.10.9.12 2.11.3.6.7.116 2 4.6.7.8.12.11 1.10.5.9.2.316 2 4.6.7.12.9.10 2.8.11.1.5.316 2 4.6.8.11.10.9 5.12.2.1.3.716 2 5.6.7.8.12.10 1.9.11.3.2.416 2 5.7.6.10.11.9 12.8.3.2.1.417 1 1.6.3.9.8.12 5.10.11.7.4.217 1 1.6.3.10.8.11 5.12.9.7.4.217 1 1.5.4.10.7.12 6.11.9.8.3.217 1 1.4.6.10.7.11 8.12.9.2.5.317 1 1.4.6.9.8.11 7.12.10.3.5.217 1 1.4.8.7.9.10 11.12.6.5.3.217 1 1.5.6.12.7.8 9.11.10.2.3.417 1 1.5.6.7.11.9 10.12.8.4.3.217 1 1.5.6.9.10.8 12.11.7.4.3.217 1 2.5.3.10.7.12 8.11.9.6.4.117 1 2.3.8.5.11.10 9.12.7.6.4.117 1 2.6.3.7.9.12 4.11.8.10.5.117 1 2.3.6.10.7.11 8.12.9.4.5.117 1 2.3.7.9.8.10 6.12.11.4.5.117 1 2.4.6.12.5.10 8.11.9.3.1.717 1 2.4.5.8.11.9 6.12.10.7.3.117 1 2.4.6.7.12.8 9.11.10.5.3.117 1 2.4.6.8.10.9 7.11.12.5.3.117 1 2.5.6.8.11.7 12.10.9.4.3.117 1 3.4.5.6.12.9 7.11.8.10.2.117 1 3.4.6.5.10.11 7.12.8.9.2.117 1 3.4.5.7.12.8 6.11.10.9.1.217 1 3.4.5.7.11.9 6.10.12.8.1.217 1 3.4.6.7.11.8 9.10.12.5.1.217 1 3.4.6.7.10.9 5.11.12.8.2.117 1 3.5.7.6.10.8 11.12.9.4.2.117 2 4.8.9.11.10.12 7.5.2.1.6.317 2 5.9.10.7.11.12 4.3.8.6.1.217 2 6.7.11.8.10.12 1.4.5.2.9.318 1 1.10.4.11.7.12 9.8.6.3.5.218 1 1.8.9.11.4.12 5.10.3.2.6.718 1 1.6.9.10.7.12 8.11.5.3.2.418 1 2.8.9.11.3.12 4.10.6.1.5.7

69

18 1 2.7.11.4.9.12 6.10.3.8.5.118 1 2.6.5.11.9.12 7.10.8.4.3.118 1 2.5.8.9.11.10 6.12.7.4.3.118 1 2.6.7.10.8.12 4.11.9.3.5.118 1 3.6.4.11.9.12 5.10.8.7.1.218 1 3.4.7.10.9.12 5.11.8.6.2.118 1 3.4.8.9.11.10 7.12.6.5.1.218 1 3.5.6.11.8.12 7.10.9.2.4.118 1 3.5.6.10.9.12 8.11.7.4.2.118 1 3.5.7.10.12.8 9.11.6.4.1.218 1 3.6.7.8.12.9 10.11.5.4.1.218 1 3.6.7.8.11.10 5.12.9.4.1.218 1 4.5.6.12.7.11 8.9.10.1.3.218 1 4.5.6.10.8.12 3.9.11.7.2.118 1 4.5.6.10.9.11 3.12.8.7.1.218 1 4.5.7.8.12.9 6.11.10.3.1.218 1 4.5.7.11.8.10 9.12.6.1.3.218 1 4.6.7.9.8.11 3.12.10.5.2.119 1 3.9.6.10.11.12 4.8.7.5.2.119 1 3.7.8.11.10.12 5.9.6.4.1.219 1 4.6.8.12.10.11 5.9.7.2.1.319 1 5.6.7.11.10.12 2.9.8.4.3.119 1 5.6.9.8.11.12 2.10.7.3.4.119 1 5.7.8.9.12.10 4.11.6.3.1.220 1 7.8.9.10.12.11 2.6.5.4.3.1

V,E=7Vertex-AntiMagic Labelings for Cycle with v,e=7.

a d E={e1,e2,e3,e4,e5,e6,e7} V={v1,v2,v3,v4,v5,v6,v7}

6 5 1.2.4.6.13.9.7 8.3.5.11.12.14.107 5 1.2.3.8.13.10.12 9.4.7.6.11.14.57 5 1.2.3.9.11.10.13 8.4.7.5.12.6.148 4 1.2.3.4.6.8.11 12.5.7.9.10.14.138 5 1.3.6.12.13.11.10 2.4.9.5.8.14.79 4 1.2.3.4.12.7.13 11.6.8.10.5.14.99 4 1.2.3.9.10.4.13 7.6.8.5.14.11.129 4 1.2.3.4.12.11.9 7.6.8.14.13.10.59 4 1.2.3.5.6.12.13 7.10.4.9.14.11.89 4 1.2.3.5.9.10.12 4.6.8.13.11.14.79 4 1.2.4.5.6.11.13 7.10.3.8.14.12.99 4 1.2.4.5.7.12.11 9.10.3.8.13.14.69 4 1.2.4.8.12.10.5 3.14.7.13.9.11.69 4 1.2.4.5.9.10.11 13.6.7.12.3.14.89 4 1.3.4.6.9.12.7 5.13.2.11.10.8.1410 4 1.2.7.13.10.12.4 5.11.9.14.3.8.610 4 1.2.5.6.10.12.13 8.7.11.3.14.4.9

70

10 4 1.2.6.8.13.9.10 3.7.14.4.5.12.1110 4 1.3.4.12.5.11.13 8.6.7.2.9.14.1010 4 1.3.4.12.9.13.7 2.10.11.6.5.8.1410 4 1.3.5.7.8.12.13 4.6.14.2.11.10.910 4 1.3.5.8.12.11.9 4.6.10.13.2.7.1410 4 2.3.6.8.10.9.11 1.5.13.4.12.7.1410 4 2.4.5.7.12.9.10 6.8.1.14.3.13.1111 3 1.2.3.4.5.7.13 12.11.6.10.14.8.911 3 1.2.3.4.5.8.12 7.11.6.10.14.13.911 3 1.2.3.4.5.11.9 10.8.12.7.14.13.611 3 1.2.4.5.6.7.10 3.8.11.14.9.13.1211 3 2.3.4.5.6.7.8 1.9.10.11.12.13.1411 3 2.3.4.5.6.7.8 1.9.10.11.12.13.1411 4 1.3.9.8.13.12.10 4.7.11.2.14.6.511 4 2.3.7.11.8.13.12 5.10.1.9.4.14.611 4 2.4.6.8.12.13.11 14.5.9.1.3.10.712 3 1.2.3.4.13.8.11 6.12.7.14.10.9.512 3 1.2.5.3.7.11.13 10.12.14.4.8.9.612 3 1.2.3.10.5.8.13 4.12.7.11.6.14.912 3 1.2.3.5.8.12.11 6.9.10.13.14.4.712 3 1.2.3.7.6.11.12 5.9.10.14.8.13.412 3 1.2.3.8.6.10.12 11.9.13.4.7.14.512 3 1.2.4.10.6.8.11 3.9.12.7.14.13.512 3 1.2.5.6.7.13.8 3.12.11.10.14.4.912 3 1.3.4.5.6.11.12 2.8.14.9.13.10.712 3 1.3.4.5.13.7.9 2.14.8.12.6.10.1112 3 1.3.4.5.7.10.12 14.8.11.6.9.13.212 3 1.3.4.6.7.8.13 10.11.14.2.5.12.912 3 1.3.5.6.7.9.11 12.8.13.4.14.2.1012 3 2.3.4.5.6.10.12 1.7.14.9.13.11.812 3 2.3.4.7.5.8.13 6.10.11.1.12.14.912 3 2.3.4.6.7.8.12 1.13.5.11.14.9.1012 3 3.4.5.6.7.9.8 1.11.12.13.2.14.1013 3 1.2.3.7.13.11.12 9.10.14.6.5.4.813 3 1.2.6.4.13.11.12 3.10.14.9.8.7.513 3 1.2.8.9.12.4.13 11.10.6.5.7.3.1413 3 1.2.4.10.9.11.12 6.13.7.14.3.5.813 3 1.2.9.5.7.12.13 11.10.8.14.4.3.613 3 1.2.5.11.13.9.8 7.10.12.6.4.3.1413 3 1.2.5.12.10.8.11 7.13.6.14.3.4.913 3 1.2.6.7.12.8.13 5.10.14.3.9.11.413 3 1.2.6.11.7.9.13 5.10.8.14.4.12.313 3 1.3.4.8.9.11.13 14.12.6.10.2.5.713 3 1.3.5.6.10.11.13 2.9.14.8.12.4.713 3 1.3.5.7.10.12.11 4.9.14.13.2.6.813 3 1.4.5.6.8.13.12 9.14.7.2.11.10.313 3 1.4.5.6.13.9.11 10.14.7.2.12.3.813 3 1.4.5.6.10.11.12 3.14.13.2.9.7.813 3 1.4.5.9.7.11.12 3.8.10.14.6.13.213 3 1.4.6.8.9.10.11 13.14.3.2.5.12.7

71

13 3 2.3.4.10.5.13.12 11.8.9.14.7.1.613 3 2.3.4.6.11.13.10 1.14.9.12.8.7.513 3 2.3.4.7.12.8.13 1.14.6.11.9.5.1013 3 2.3.4.13.11.7.9 5.8.12.14.1.10.613 3 2.3.4.7.11.10.12 5.8.9.14.13.1.613 3 2.3.4.8.9.10.13 7.11.12.1.14.6.513 3 2.3.5.7.9.10.13 4.11.14.1.12.6.813 3 4.5.6.7.9.8.10 11.13.2.3.12.14.114 2 1.2.3.4.6.12.7 14.11.13.9.10.8.514 3 1.2.7.10.12.11.13 9.14.5.3.4.6.814 3 1.2.8.9.11.13.12 10.14.4.3.6.5.714 3 1.3.9.12.7.11.13 6.10.5.8.4.14.214 3 1.4.5.10.11.12.13 6.9.8.14.2.3.714 3 1.5.6.8.11.13.12 4.14.3.9.10.2.714 3 1.5.7.8.12.10.13 6.11.2.14.3.4.914 3 1.5.7.9.11.10.13 3.8.14.4.12.2.614 3 1.5.8.9.10.12.11 2.14.4.6.13.7.314 3 2.4.5.9.13.12.11 7.8.14.3.10.1.614 3 2.5.6.7.11.12.13 8.10.9.1.14.3.414 3 2.5.6.8.10.12.13 11.7.9.3.14.1.414 3 2.5.6.9.10.13.11 4.7.12.14.1.3.814 3 2.6.7.8.13.9.11 1.12.4.14.5.10.314 3 3.4.5.8.12.13.11 6.10.14.1.9.7.214 3 4.5.6.7.10.13.11 2.14.9.1.12.3.814 3 5.6.7.8.10.9.11 1.3.13.14.2.4.1215 2 1.2.3.6.10.7.13 9.12.14.8.11.4.515 2 1.2.3.8.6.9.13 7.14.10.12.11.4.515 2 1.2.3.6.10.9.11 13.12.14.8.5.4.715 2 1.2.3.7.8.12.9 11.14.10.13.4.5.615 2 1.2.3.7.8.10.11 5.12.14.13.6.9.415 2 1.2.4.7.10.5.13 11.14.9.8.6.12.315 2 1.2.4.5.8.9.13 7.12.11.10.14.6.315 2 1.2.4.6.10.12.7 11.14.9.13.5.3.815 2 1.2.4.6.12.8.9 5.14.13.11.7.3.1015 2 1.2.4.7.8.9.11 3.14.13.10.12.6.515 2 1.2.5.7.9.6.12 10.14.8.13.11.4.315 2 1.2.5.6.10.7.11 13.14.8.12.3.4.915 2 1.2.5.6.8.9.11 7.12.10.14.13.4.315 2 1.2.5.7.9.8.10 4.14.12.13.11.6.315 2 1.3.4.5.10.6.13 7.11.12.14.2.9.815 2 1.3.4.5.9.8.12 2.13.14.10.11.6.715 2 1.3.4.5.8.10.11 7.13.14.6.12.9.215 2 1.3.4.6.7.10.11 9.13.12.5.14.8.215 2 2.3.4.5.9.11.8 7.10.12.14.13.1.615 2 2.3.5.6.8.7.11 4.10.13.14.9.12.115 2 2.3.5.6.7.9.10 11.14.13.4.12.1.815 3 4.6.7.10.11.13.12 5.14.2.1.9.3.815 3 5.6.7.9.11.12.13 3.4.14.2.10.1.815 3 6.7.8.9.10.12.11 4.2.3.13.14.5.116 1 1.2.3.4.5.6.7 8.14.13.12.11.10.9

72

16 2 1.5.2.10.7.11.13 8.12.9.14.3.6.416 2 1.2.6.7.12.10.11 8.13.14.5.9.4.316 2 1.2.6.8.9.11.12 7.13.10.14.5.4.316 2 1.2.7.8.10.12.9 14.13.11.3.4.6.516 2 1.3.4.10.13.6.12 11.14.9.8.5.7.216 2 1.3.4.10.7.13.11 12.14.9.6.5.8.216 2 1.3.4.10.8.11.12 9.14.13.2.6.7.516 2 1.3.5.6.12.9.13 10.14.8.11.2.7.416 2 1.3.5.7.9.11.13 2.14.12.10.8.6.416 2 1.3.5.8.10.9.13 2.14.12.11.4.7.616 2 1.3.6.8.7.11.13 10.12.9.14.5.4.216 2 1.3.6.10.7.9.13 14.12.11.8.5.2.416 2 1.3.6.12.7.9.11 4.14.13.2.5.10.816 2 1.3.6.8.9.10.12 7.14.13.2.11.5.416 2 1.3.7.8.10.9.11 4.14.12.13.2.5.616 2 1.4.6.7.8.12.11 14.13.10.3.9.2.516 2 2.3.5.6.8.12.13 7.11.10.9.14.4.116 2 2.3.5.7.9.11.12 6.13.14.4.10.8.116 2 2.3.5.8.9.10.12 4.11.14.13.7.1.616 2 2.4.5.6.7.13.12 10.14.9.11.3.8.116 2 2.4.5.7.10.13.8 6.14.9.12.11.3.116 2 2.4.5.7.8.11.12 6.10.9.14.13.3.116 2 2.4.6.7.9.8.13 1.14.12.5.10.11.316 2 2.4.6.7.9.10.11 3.14.8.13.12.5.116 2 3.4.5.7.8.10.12 1.11.13.14.9.2.616 3 7.8.9.10.11.12.13 14.1.2.3.4.5.617 1 1.2.3.4.9.6.10 11.14.13.12.7.8.517 1 1.2.3.4.7.8.10 9.14.13.12.11.6.517 1 1.2.3.5.6.7.11 9.14.13.12.8.10.417 1 1.2.3.5.6.8.10 9.14.13.11.12.7.417 1 1.2.3.5.7.8.9 11.14.13.12.10.4.617 1 1.2.4.5.6.8.9 13.14.12.10.11.7.317 1 1.3.4.5.6.7.9 10.13.12.14.11.8.217 2 1.4.6.9.11.13.12 8.14.7.10.3.5.217 2 1.4.7.8.12.11.13 9.14.10.2.5.6.317 2 1.4.7.13.10.12.9 11.14.6.3.2.5.817 2 1.4.8.9.10.11.13 7.14.5.12.6.2.317 2 1.5.6.9.10.12.13 7.11.8.14.4.3.217 2 1.6.7.8.10.11.13 3.14.12.4.9.2.517 2 1.6.7.9.11.12.10 8.14.4.13.3.2.517 2 2.3.5.10.11.13.12 7.14.9.8.6.1.417 2 2.3.6.9.13.11.12 7.14.8.10.1.5.417 2 2.3.7.9.10.12.13 8.14.11.1.6.5.417 2 2.3.8.9.11.13.10 5.14.12.4.7.1.617 2 2.5.7.8.12.9.13 4.10.11.14.1.6.317 2 3.4.6.8.10.12.13 1.14.9.11.5.7.217 2 3.4.6.9.10.11.13 5.12.7.14.8.2.117 2 3.4.7.8.10.11.13 1.12.14.6.9.2.517 2 3.5.6.7.10.12.13 11.9.14.8.2.1.417 2 3.5.6.8.9.12.13 1.11.14.7.10.2.4

73

17 2 3.5.7.8.10.11.12 6.9.13.14.1.2.417 2 4.5.6.8.9.11.13 2.14.10.3.12.7.117 2 4.5.7.8.10.13.9 14.12.11.2.1.6.317 2 4.5.7.8.9.11.12 3.14.13.2.10.1.618 1 1.3.4.6.8.11.9 10.14.12.13.7.5.218 1 1.3.4.7.8.9.10 11.14.12.13.5.6.218 1 1.3.5.6.7.8.12 10.14.13.9.11.4.218 1 1.3.5.6.8.9.10 11.14.12.13.7.2.418 1 1.4.5.6.7.8.11 12.14.13.9.10.3.218 1 1.4.5.6.7.9.10 12.14.11.13.8.2.318 1 2.3.4.5.9.6.13 7.14.11.12.10.8.118 1 2.3.4.5.7.9.12 6.13.14.10.11.8.118 1 2.3.4.5.7.10.11 8.14.13.9.12.6.118 1 2.3.4.6.7.9.11 5.14.13.12.10.8.118 1 2.3.4.6.8.9.10 11.13.12.14.7.5.118 1 2.4.5.6.7.8.10 9.14.13.12.11.3.118 2 1.7.9.10.12.11.13 6.14.2.5.8.3.418 2 2.7.8.10.11.13.12 14.9.5.4.3.6.118 2 4.6.8.9.11.12.13 1.10.14.7.2.3.518 2 5.6.8.9.10.12.13 4.7.14.3.11.2.119 1 1.4.8.10.11.2.13 6.14.9.5.3.12.719 1 1.4.5.7.11.8.13 9.14.12.10.2.6.319 1 1.4.5.7.9.10.13 6.14.12.11.8.3.219 1 1.4.5.8.9.10.12 7.14.13.11.6.2.319 1 1.4.6.7.8.10.13 9.14.12.11.5.3.219 1 1.4.6.9.7.12.10 11.14.13.5.8.2.319 1 1.5.6.7.8.13.9 12.14.10.11.4.2.319 1 1.5.6.7.10.8.12 9.13.14.11.4.2.319 1 1.5.6.7.9.10.11 12.13.14.8.4.3.219 1 2.3.6.9.12.4.13 8.14.11.10.1.5.719 1 2.3.4.8.9.11.12 7.14.13.10.6.5.119 1 2.3.5.6.9.11.13 7.14.12.10.8.4.119 1 2.3.5.6.10.11.12 8.14.13.9.7.4.119 1 2.3.5.7.8.11.13 6.14.12.10.9.4.119 1 2.3.6.7.8.13.10 11.14.12.9.5.4.119 1 2.3.6.11.8.7.12 9.14.13.4.1.10.519 1 2.3.6.7.9.10.12 11.14.13.8.4.5.119 1 2.4.5.6.8.11.13 9.14.10.12.7.3.119 1 2.4.5.6.9.13.10 7.14.12.11.8.3.119 1 2.4.5.6.9.11.12 7.14.13.8.10.3.119 1 2.4.5.8.9.11.10 7.14.13.12.6.1.319 1 2.4.6.7.8.10.12 11.13.14.9.5.3.119 1 2.5.6.7.8.9.12 11.14.13.10.4.3.119 1 2.5.6.7.8.11.10 13.14.12.9.4.1.319 1 3.4.5.6.7.11.13 8.12.14.10.9.2.119 1 3.4.5.6.8.10.13 9.12.14.11.7.2.119 1 3.4.5.6.8.11.12 7.13.10.14.9.2.119 1 3.4.5.6.10.12.9 13.14.11.8.7.2.119 1 3.4.5.7.8.9.13 6.14.11.12.10.2.119 1 3.4.5.7.9.10.11 6.12.14.13.8.2.1

74

19 1 3.4.6.7.8.9.12 5.14.13.11.10.2.119 1 3.4.6.7.8.10.11 5.14.13.12.9.2.119 1 3.5.6.7.8.11.9 12.13.14.10.4.1.220 1 1.6.7.8.12.9.13 11.14.10.5.2.3.420 1 2.4.6.9.10.12.13 7.14.11.8.5.3.120 1 2.4.7.8.10.12.13 6.14.11.9.5.3.120 1 2.4.7.9.10.11.13 6.14.12.8.3.5.120 1 2.4.8.9.10.11.12 7.14.13.6.3.5.120 1 2.5.7.10.13.8.11 9.14.12.6.3.4.120 1 2.5.7.9.10.11.12 6.14.13.8.4.1.320 1 2.6.7.8.10.11.12 9.14.13.5.3.4.120 1 3.4.5.9.10.12.13 7.14.11.8.6.2.120 1 3.4.6.7.11.12.13 8.14.10.9.5.2.120 1 3.4.7.8.9.13.12 5.14.11.10.6.2.120 1 3.4.7.9.10.11.12 8.13.14.6.2.5.120 1 3.5.6.8.10.13.11 7.12.14.9.4.1.220 1 3.5.6.9.10.12.11 7.14.13.8.1.4.220 1 3.5.7.8.9.11.13 4.14.12.10.6.1.220 1 3.6.7.8.9.10.13 4.14.12.11.5.2.120 1 3.6.7.8.9.11.12 5.14.13.10.4.2.120 1 4.5.6.7.12.13.9 8.11.14.10.3.1.220 1 4.5.6.8.10.11.12 9.14.13.7.2.1.320 1 4.6.7.8.12.9.10 11.14.13.5.1.2.321 1 2.6.9.10.11.13.12 7.14.8.5.4.3.121 1 3.5.9.10.11.12.13 8.14.7.4.6.2.121 1 3.6.8.10.11.12.13 9.14.7.4.5.1.221 1 3.7.8.9.11.12.13 5.14.10.6.2.4.121 1 4.5.8.10.11.12.13 7.14.9.3.6.2.121 1 4.7.8.9.10.13.12 5.14.11.6.3.1.221 1 5.7.8.9.10.11.13 4.14.12.6.2.3.1

V,E=8 for a=6 d=5 onlyVertex-AntiMagic Labelings for Cycle with v,e=7.

a d E={e1,e2,e3,e4,e5,e6,e7} V={v1,v2,v3,v4,v5,v6,v7}

6 5 1.2.5.8.11.14.7.4 6.3.9.13.12.16.15.10

B2: (a,d)-VERTEX-ANTIMAGIC TOTAL LABELINGS FOR PATHS

V1 V2 V3 V4

E1 E2 E3

75

V=4, E=3Vertex-AntiMagic Labelings for Path with v=4, e=3.

a d E={e1,e2,e3} V={v1,v2,v3,v4}

3 5 2.7.5 1.4.6.35 3 1.7.2 4.3.5.65 3 1.3.6 4.7.5.26 2 1.5.2 7.6.3.46 2 1.3.4 7.2.5.66 3 1.7.6 5.4.2.36 3 3.7.4 6.5.1.27 1 1.2.3 6.5.4.77 2 1.7.4 6.3.2.57 2 5.1.6 2.3.4.77 2 2.3.7 5.4.1.67 2 2.4.6 5.3.1.77 2 4.3.5 7.6.1.27 3 5.7.6 2.1.3.48 1 1.5.4 7.3.2.68 1 2.5.3 6.4.1.78 2 3.7.6 5.2.1.48 2 4.5.7 6.3.2.19 1 3.6.5 7.2.1.4

V=5, E=4Vertex-AntiMagic Labelings for Path with v=5, e=4.

a d E={e1,e2,e3,e4} V={v1,v2,v3,v4,v5}

3 5 1.4.9.6 2.3.5.8.75 4 2.1.9.8 3.6.7.4.55 4 1.5.6.8 4.3.2.7.95 4 2.5.9.4 7.6.3.8.16 3 1.3.2.9 5.8.4.7.66 3 1.2.4.8 5.9.3.6.76 3 1.3.5.6 8.2.4.7.96 3 2.3.4.6 7.1.5.8.96 4 1.9.7.8 5.4.6.3.26 4 4.7.5.9 2.3.6.8.17 3 2.3.6.9 5.8.1.4.78 3 3.6.7.9 5.2.1.4.89 1 1.2.3.4 8.7.6.5.99 2 4.7.1.8 5.2.3.6.99 2 2.4.6.8 7.5.3.1.99 2 2.5.7.6 9.8.1.4.3

76

9 3 6.7.9.8 3.2.5.1.410 1 1.5.2.7 9.8.4.3.610 1 1.7.4.3 9.6.2.5.810 2 3.5.9.8 7.4.2.1.611 1 3.7.1.9 8.2.5.4.611 1 2.7.3.8 9.4.5.1.611 1 3.4.5.8 9.6.2.1.711 1 3.4.7.6 8.5.2.1.912 1 6.8.2.9 7.1.4.5.312 1 4.8.6.7 9.2.1.3.5

V=6, E=5Vertex-AntiMagic Labelings for Path with v=6, e=5.

a d E={e1,e2,e3,e4,e5} V={v1,v2,v3,v4,v5,v6}

3 5 1.3.11.7.5 2.4.9.10.6.83 5 1.4.5.10.7 2.3.9.8.11.64 5 1.2.10.11.9 3.6.7.8.4.54 5 1.4.10.11.7 3.9.5.8.6.24 5 1.11.8.9.4 3.2.10.7.6.54 5 3.2.7.11.10 1.4.5.6.8.95 5 4.6.11.10.8 1.5.3.9.7.26 4 1.6.11.4.8 5.3.9.7.2.106 4 4.1.6.9.10 2.5.7.11.3.86 4 1.10.7.4.8 5.3.9.11.6.26 4 2.5.3.11.9 4.7.10.8.6.16 4 2.3.6.11.8 4.9.1.5.7.106 4 2.5.6.7.10 4.3.11.1.9.86 4 3.10.5.8.4 7.1.11.9.6.26 4 3.6.7.10.4 11.1.5.9.8.26 4 3.6.9.8.4 7.5.11.1.10.26 4 4.5.6.7.8 2.1.3.9.11.107 3 1.2.3.4.11 6.10.5.9.7.87 3 1.2.3.5.10 9.4.8.11.7.67 3 1.2.3.8.7 6.10.5.11.4.97 4 1.10.5.11.9 6.4.8.3.7.27 4 1.8.9.7.11 6.2.10.3.5.47 4 2.9.11.10.4 5.8.3.6.1.77 4 2.6.8.11.9 5.3.1.4.7.107 4 2.7.9.10.8 5.6.11.4.1.37 4 3.11.5.8.9 4.1.7.6.10.27 4 3.7.11.6.9 4.5.1.10.8.27 4 4.6.9.7.10 3.1.8.11.2.58 3 1.11.3.2.10 7.5.9.6.8.48 3 1.2.4.9.11 10.5.8.7.3.68 3 1.2.10.9.5 7.11.8.4.3.68 3 1.2.6.10.8 7.11.3.4.5.9

77

8 3 1.8.3.11.4 7.2.6.9.5.108 3 1.4.5.11.6 10.9.8.7.3.28 3 1.4.8.5.9 7.6.2.10.3.118 3 1.8.6.7.5 10.11.9.4.2.38 3 2.4.3.7.11 6.8.10.1.5.99 2 1.3.4.2.8 10.5.6.9.7.119 2 1.3.5.2.7 8.9.11.4.6.109 2 1.5.2.4.6 10.3.8.7.9.119 3 3.1.10.8.11 9.5.4.6.2.79 3 1.5.6.11.10 8.9.7.4.3.29 3 1.5.7.9.11 8.6.3.2.4.109 3 1.5.9.10.8 11.3.4.2.6.79 3 6.1.7.8.11 3.5.10.9.2.49 3 1.6.7.10.9 8.11.2.4.5.39 3 2.3.8.11.9 10.4.7.5.1.69 3 2.4.6.10.11 7.9.8.5.3.19 3 2.4.9.11.7 10.3.5.1.6.89 3 2.4.8.9.10 7.6.3.1.5.119 3 2.5.6.9.11 7.8.1.3.4.109 3 6.2.8.7.10 3.4.5.9.1.119 3 3.4.10.5.11 6.8.7.9.2.19 3 3.7.4.11.8 6.5.1.9.2.109 3 3.4.7.9.10 6.8.1.2.5.119 3 3.5.6.8.11 9.1.4.10.2.79 3 3.5.8.7.10 6.4.2.9.1.1110 2 1.7.3.11.2 9.8.4.6.5.1010 2 1.2.8.3.10 11.7.4.9.5.610 2 1.4.2.6.11 9.7.8.10.3.510 2 1.2.4.7.10 9.11.6.5.3.810 2 1.2.4.8.9 11.7.10.6.3.510 2 1.2.5.6.10 11.7.9.3.4.810 2 1.5.7.9.2 11.10.6.4.3.810 2 1.2.6.7.8 9.11.4.3.5.1010 2 1.3.11.5.4 9.8.6.2.7.1010 2 1.4.3.10.6 9.11.5.7.2.810 2 1.3.9.7.4 11.6.8.2.5.1010 2 1.3.9.5.6 11.10.8.2.7.410 2 1.5.7.3.8 11.4.2.6.9.1010 2 1.4.5.6.8 9.7.11.3.2.1010 2 2.3.4.10.5 8.7.11.6.1.910 2 2.3.4.6.9 8.7.11.10.1.510 2 2.3.4.7.8 10.5.11.9.1.610 2 2.3.6.8.5 10.9.1.4.7.1110 2 2.4.5.6.7 8.10.3.9.1.1110 3 8.1.11.9.10 2.4.7.5.3.610 3 2.7.9.10.11 8.4.3.6.1.510 3 5.11.4.10.9 8.3.7.2.6.110 3 4.7.8.11.9 6.2.1.3.5.1010 3 5.10.6.11.7 8.1.9.2.4.3

78

10 3 5.7.9.10.8 11.1.3.6.4.211 1 1.2.3.4.5 10.9.8.7.6.1111 2 2.6.11.1.10 9.5.4.3.8.711 2 1.7.9.2.11 10.5.3.4.8.611 2 2.10.1.9.8 11.7.6.5.4.311 2 3.5.10.1.11 8.7.4.2.9.611 2 1.6.9.3.11 10.8.4.5.7.211 2 1.3.7.8.11 10.9.5.4.2.611 2 3.1.10.7.9 8.11.6.2.5.411 2 1.4.9.5.11 10.8.2.7.3.611 2 1.5.6.7.11 10.9.2.4.3.811 2 6.1.5.8.10 7.4.11.2.3.911 2 1.5.7.8.9 10.11.3.6.2.411 2 2.3.11.4.10 9.8.1.6.5.711 2 2.6.9.3.10 11.7.4.5.8.111 2 2.3.7.10.8 11.6.5.4.1.911 2 2.4.8.5.11 9.7.3.6.1.1011 2 2.5.9.4.10 11.8.3.6.7.111 2 4.6.2.7.11 9.5.3.8.1.1011 2 2.4.6.8.10 9.7.5.3.1.1111 2 2.4.8.7.9 11.5.3.6.1.1011 2 2.5.9.6.8 11.10.1.4.7.311 2 3.4.5.7.11 8.10.6.9.1.211 2 3.5.6.7.9 8.11.10.2.1.411 3 7.8.9.10.11 4.2.6.1.5.312 1 1.2.6.3.9 11.10.7.5.4.812 1 1.2.4.5.9 11.10.8.6.3.712 1 1.4.2.6.8 11.9.10.5.3.712 1 1.2.5.6.7 11.10.8.3.4.912 1 1.3.4.5.8 11.9.10.7.2.612 1 1.3.4.7.6 11.9.10.5.2.812 1 2.3.4.7.5 10.9.8.6.1.1112 2 8.1.10.6.11 4.7.3.2.5.912 2 5.2.10.8.11 7.9.6.4.1.312 2 6.2.7.10.11 8.4.9.3.1.512 2 3.4.8.10.11 9.7.6.2.1.512 2 3.5.7.11.10 9.6.4.2.1.812 2 3.6.8.9.10 11.7.4.5.1.212 2 4.5.6.10.11 8.7.3.2.1.912 2 4.9.5.7.11 10.3.8.6.2.112 2 5.4.8.9.10 7.11.2.1.3.612 2 4.6.7.8.11 10.2.5.1.3.912 2 5.6.7.8.10 11.1.9.3.2.413 1 5.1.8.2.11 9.10.6.3.4.713 1 6.1.7.2.11 8.10.5.9.3.413 1 2.9.1.8.7 11.4.6.5.3.1013 1 3.4.9.1.10 11.6.2.7.5.813 1 3.5.8.1.10 11.7.4.9.2.613 1 3.1.7.6.10 11.9.8.4.2.5

79

13 1 3.1.8.6.9 10.11.5.4.2.713 1 4.1.6.7.9 11.8.10.3.2.513 1 2.3.8.4.10 11.9.5.6.1.713 1 3.2.6.5.11 10.9.8.4.1.713 1 2.3.10.5.7 11.9.4.1.6.813 1 2.3.5.8.9 11.10.6.4.1.713 1 2.3.7.6.9 11.10.4.5.1.813 1 2.4.5.6.10 11.9.7.3.1.813 1 2.4.7.5.9 11.10.3.6.1.813 1 2.4.6.7.8 11.10.5.1.3.913 1 3.4.5.9.6 10.11.7.1.2.813 1 3.4.5.7.8 10.11.6.2.1.913 2 8.4.9.11.10 5.3.6.1.2.713 2 5.7.10.9.11 8.3.2.4.1.613 2 6.7.11.8.10 9.4.1.2.5.314 1 5.2.8.7.11 9.10.6.3.1.414 1 5.2.9.7.10 11.8.6.3.1.414 1 4.3.9.6.11 10.8.5.1.2.714 1 3.5.9.6.10 11.7.2.4.1.814 1 3.7.6.8.9 11.5.4.2.1.1014 1 4.5.11.6.7 10.8.3.1.2.914 1 4.5.6.8.10 11.9.3.2.1.714 1 4.5.8.9.7 10.6.3.2.1.1115 1 9.3.10.6.11 7.5.2.4.1.815 1 7.8.10.3.11 9.2.1.5.6.415 1 5.8.9.6.11 10.4.2.1.3.7

V=7, E=6Vertex-AntiMagic Labelings for Path with v=7, e=6.

a d E={e1,e2,e3,e4,e5,e6} V={v1,v2,v3,v4,v5,v6,v7}

3 5 1.3.5.12.8.6 2.4.10.11.13.9.74 5 1.2.4.10.12.13 3.6.8.5.7.9.114 5 1.2.7.12.9.11 3.6.5.10.13.4.84 5 1.2.8.10.9.12 3.6.4.11.5.13.75 4 1.2.3.5.7.10 4.6.8.9.13.12.115 5 1.3.9.11.12.13 4.6.8.5.7.10.25 5 2.5.11.12.10.9 3.8.4.7.13.6.15 5 3.10.13.8.11.4 7.2.12.9.6.5.16 4 1.2.3.11.6.12 5.7.9.4.13.8.106 4 1.2.8.9.3.12 5.7.4.13.6.11.106 4 1.2.3.11.10.8 5.7.13.12.9.4.66 4 1.2.4.5.11.12 9.3.8.13.10.7.66 4 1.2.4.8.9.11 5.7.12.10.13.6.36 4 1.2.5.13.6.8 9.3.7.4.11.12.106 4 1.3.4.5.10.12 9.2.7.13.11.8.66 4 1.3.4.6.13.8 5.10.11.12.7.9.2

80

6 4 1.3.4.6.11.10 9.2.7.8.13.5.126 4 1.3.7.11.9.4 5.6.8.12.2.13.106 4 1.3.4.8.9.10 5.6.11.2.13.7.126 4 1.3.6.7.8.10 9.2.13.5.11.12.46 4 2.3.6.4.13.7 8.1.5.12.9.10.116 4 2.3.5.8.11.6 12.1.10.9.7.13.47 4 1.2.5.11.13.10 6.12.4.7.3.8.97 4 1.2.8.11.7.13 10.4.5.12.9.3.67 4 1.2.9.7.13.10 6.8.12.3.11.4.57 4 1.3.4.13.9.12 6.7.8.2.5.10.117 4 1.3.8.12.5.13 6.7.4.11.2.9.107 4 1.6.12.13.7.3 10.8.5.2.11.9.47 4 1.6.12.9.11.3 10.8.13.2.7.5.47 4 1.4.8.5.13.11 10.2.3.6.9.7.127 4 1.4.5.9.11.12 6.10.2.13.3.8.77 4 1.5.7.12.8.9 6.13.3.4.11.10.27 4 2.3.4.11.13.9 5.6.8.12.7.1.107 4 2.3.11.4.10.12 5.6.1.8.13.9.77 4 2.5.8.13.11.3 9.12.10.6.7.1.47 4 2.3.11.8.12.6 9.10.5.4.7.13.17 4 2.4.6.10.13.7 5.9.1.11.8.3.127 4 2.4.6.7.11.12 5.13.1.10.9.8.37 4 2.4.7.11.10.8 5.9.12.1.6.13.38 3 1.2.3.5.4.13 7.8.12.6.11.9.108 3 1.2.3.4.6.12 10.5.9.13.7.8.118 3 1.2.3.4.7.11 10.5.9.13.6.8.128 3 1.2.3.4.10.8 7.11.6.13.12.5.98 3 1.3.4.5.6.9 7.10.13.2.12.11.88 4 1.2.13.11.10.12 7.9.5.8.3.6.48 4 1.4.10.9.13.12 11.3.2.5.6.7.88 4 1.5.8.10.13.12 7.6.11.2.9.3.48 4 1.7.12.13.10.6 11.8.5.3.9.4.28 4 1.6.8.12.13.9 7.5.2.4.3.10.118 4 1.13.10.6.11.8 7.2.9.12.3.5.48 4 2.8.3.12.13.11 6.10.1.9.7.4.58 4 2.3.13.10.9.12 6.7.8.5.1.11.48 4 2.5.6.11.13.12 10.1.9.7.8.3.48 4 2.5.8.9.13.12 10.1.11.3.6.7.48 4 2.6.9.12.13.7 10.8.5.11.3.4.18 4 2.8.7.12.11.9 6.10.1.13.5.4.38 4 3.4.6.12.11.13 5.9.2.10.1.8.78 4 4.5.9.13.11.7 12.3.6.2.8.10.19 3 1.2.3.12.4.13 8.9.10.6.11.7.59 3 1.2.13.5.3.11 8.9.12.6.7.4.109 3 1.2.3.6.10.13 8.12.7.9.5.4.119 3 1.2.3.7.13.9 11.6.10.8.4.5.129 3 1.2.3.12.7.10 11.6.13.9.8.4.59 3 1.2.4.10.5.13 8.12.6.7.3.9.119 3 1.2.6.4.13.9 8.12.10.11.7.5.3

81

9 3 1.4.2.6.10.12 11.13.3.7.8.5.99 3 1.2.4.7.8.13 11.6.9.10.12.3.59 3 1.2.9.4.7.12 11.6.10.5.13.8.39 3 1.2.4.7.11.10 8.9.12.13.3.6.59 3 1.2.6.5.10.11 8.9.13.4.12.3.79 3 1.2.7.5.9.11 8.12.3.6.13.4.109 3 1.3.9.5.7.10 8.11.6.13.12.4.29 3 1.4.5.6.12.7 8.10.3.13.9.2.119 3 2.3.4.5.10.11 7.13.8.12.9.6.19 3 2.3.4.12.6.8 13.7.11.5.9.10.19 3 2.3.4.6.9.11 7.10.5.8.12.1.139 3 2.3.5.6.7.12 10.13.1.4.11.8.99 3 2.4.5.6.8.10 7.12.3.13.1.9.119 4 2.10.8.11.13.12 7.1.3.6.9.4.59 4 3.8.9.13.11.12 6.2.4.7.1.10.59 4 4.6.10.12.11.13 5.3.1.7.2.9.89 4 5.7.12.10.13.9 4.1.2.3.6.11.810 3 1.11.12.3.2.13 9.4.5.7.8.10.610 3 2.1.5.9.12.13 11.7.10.8.4.3.610 3 1.2.6.12.8.13 9.10.11.4.5.7.310 3 1.2.6.11.13.9 12.7.8.5.4.3.1010 3 1.2.6.12.10.11 9.13.5.4.3.7.810 3 1.3.4.13.10.11 9.12.6.5.2.7.810 3 1.3.5.13.11.9 12.6.8.7.4.2.1010 3 1.5.3.12.10.11 9.13.8.7.6.4.210 3 1.7.8.11.3.12 9.5.4.6.2.13.1010 3 1.3.9.8.10.11 12.6.13.2.4.7.510 3 1.4.5.12.7.13 9.8.10.11.6.2.310 3 1.4.6.11.7.13 9.8.12.2.10.5.310 3 1.4.6.8.13.10 9.11.3.5.7.2.1210 3 1.8.4.6.11.12 9.7.10.3.2.5.1310 3 1.4.7.8.9.13 12.5.11.10.2.6.310 3 1.4.10.12.8.7 9.11.5.3.2.13.610 3 1.4.11.9.7.10 12.5.13.2.3.8.610 3 1.5.6.7.10.13 12.4.8.3.11.2.910 3 1.5.6.11.7.12 9.13.2.8.10.3.410 3 1.5.10.6.8.12 9.7.13.3.11.2.410 3 2.3.12.4.8.13 11.5.1.9.10.7.610 3 2.6.3.13.11.7 8.5.10.12.1.4.910 3 2.3.6.10.8.13 11.5.7.12.1.4.910 3 2.3.7.8.10.12 11.5.9.1.4.6.1310 3 2.4.5.6.12.13 11.10.1.8.7.3.910 3 4.2.5.11.7.13 9.10.3.6.1.8.1210 3 2.4.5.9.10.12 8.13.7.11.3.6.110 3 2.4.6.8.13.9 11.10.12.5.7.3.110 3 2.4.6.9.11.10 8.13.3.1.5.7.1210 3 3.4.5.7.12.11 13.6.1.10.9.2.810 3 3.4.5.9.8.13 7.6.10.2.11.1.1210 3 3.4.5.12.8.10 13.6.1.11.2.7.9

82

10 3 3.4.5.9.10.11 13.12.1.8.6.7.210 3 3.4.8.6.10.11 7.9.13.5.12.1.211 2 1.2.3.5.11.6 10.12.8.9.7.4.1311 2 1.3.4.5.7.8 10.9.12.6.11.2.1311 3 1.4.8.12.11.13 10.9.5.3.6.2.711 3 1.5.12.7.13.11 10.8.3.4.9.2.611 3 5.9.1.10.11.13 6.3.4.12.8.2.711 3 1.6.9.11.10.12 13.4.2.3.5.7.811 3 1.7.8.9.11.13 10.6.5.12.3.2.411 3 1.7.8.10.12.11 13.3.2.5.4.6.911 3 2.4.7.12.11.13 9.8.6.1.3.5.1011 3 2.4.10.12.8.13 9.11.6.7.3.5.111 3 2.4.9.13.10.11 12.5.7.1.3.8.611 3 2.7.5.12.10.13 9.8.11.3.4.6.111 3 2.6.7.11.10.13 9.12.1.5.8.3.411 3 2.8.11.6.10.12 9.4.7.3.13.1.511 3 3.4.9.10.11.12 8.7.13.1.2.6.511 3 3.5.11.10.7.13 8.9.4.2.12.6.111 3 3.6.7.8.12.13 11.2.4.5.9.1.1011 3 3.6.11.7.9.13 8.5.12.2.1.4.1011 3 3.6.8.10.9.13 11.2.12.5.1.7.411 3 4.5.6.9.12.13 7.11.3.2.8.1.1011 3 4.5.7.10.12.11 13.2.8.9.1.6.311 3 4.6.7.11.9.12 10.1.13.2.3.8.511 3 4.6.8.10.9.12 7.13.3.11.1.5.211 3 4.7.8.9.11.10 13.3.5.12.6.2.112 2 1.2.5.9.6.12 11.13.7.10.3.4.812 2 1.2.7.5.8.12 13.9.11.6.3.4.1012 2 1.2.5.9.8.10 11.13.7.4.3.6.1212 2 1.2.6.7.11.8 13.9.10.3.4.5.1212 2 1.2.6.7.9.10 11.13.12.5.8.3.412 2 1.3.9.5.13.4 11.10.6.8.2.7.1212 2 1.3.6.9.4.12 13.8.7.5.11.2.1012 2 1.3.4.7.8.12 11.10.9.13.5.2.612 2 1.3.5.13.7.6 11.10.8.2.4.9.1212 2 1.3.5.9.11.6 13.8.12.4.2.7.1012 2 1.3.5.11.7.8 13.12.4.6.2.9.1012 2 1.4.6.8.5.11 13.7.12.10.3.2.912 2 1.4.5.9.6.10 13.7.11.2.3.8.1212 2 1.4.5.7.8.10 11.9.13.12.3.2.612 2 1.4.6.8.7.9 13.11.12.10.5.2.312 2 2.3.4.5.13.8 10.9.11.7.6.1.1212 2 2.3.4.9.5.12 10.11.13.1.8.7.612 2 2.3.4.8.7.11 12.13.5.10.1.6.912 2 2.3.4.7.9.10 12.11.5.13.6.1.812 2 2.3.5.6.9.10 12.11.4.13.7.1.812 3 9.1.10.11.12.13 6.2.7.3.4.5.812 3 3.7.10.11.12.13 9.8.4.6.1.5.212 3 4.7.9.13.11.12 8.10.2.5.6.1.3

83

12 3 5.6.9.11.12.13 7.4.3.10.1.2.812 3 6.7.8.12.13.10 9.11.3.1.5.4.212 3 6.7.9.13.10.11 12.2.5.8.4.3.113 1 1.2.3.4.5.6 12.11.10.9.8.7.1313 2 1.6.8.11.3.13 12.10.5.4.7.9.213 2 1.3.9.6.10.13 12.11.5.4.7.2.813 2 1.3.7.8.10.13 12.11.9.6.5.2.413 2 1.5.10.4.9.13 12.11.8.7.2.3.613 2 4.1.9.6.10.12 11.8.13.2.5.3.713 2 1.5.6.8.9.13 12.11.4.7.2.3.1013 2 1.5.6.11.9.10 12.13.4.8.3.2.713 2 1.5.7.8.10.11 12.9.13.4.3.2.613 2 1.6.7.9.11.8 12.10.2.3.5.4.1313 2 2.3.5.12.7.13 11.10.9.8.4.1.613 2 2.3.5.10.9.13 11.12.7.4.6.1.813 2 2.3.9.12.5.11 13.8.7.4.6.1.1013 2 2.3.9.6.12.10 13.8.5.4.7.1.1113 2 2.3.7.8.9.13 11.12.5.10.4.1.613 2 2.3.9.7.10.11 13.12.1.5.6.4.813 2 2.4.13.5.8.10 11.9.6.1.12.3.713 2 2.4.5.11.8.12 13.7.10.1.6.3.913 2 2.4.6.8.10.12 11.9.7.5.3.1.1313 2 2.4.7.9.8.12 11.13.10.1.6.5.313 2 2.5.6.7.9.13 11.8.10.4.3.1.1213 2 2.5.7.6.10.12 11.8.9.4.3.1.1313 2 2.5.9.6.8.12 11.10.1.4.7.3.1313 2 2.5.11.6.8.10 13.12.1.4.9.7.313 2 2.5.7.8.9.11 13.12.1.10.4.3.613 2 2.6.7.9.8.10 13.11.12.1.4.5.313 2 3.4.5.11.13.6 10.12.8.7.1.2.913 2 3.4.5.7.13.10 12.6.8.11.1.2.913 2 3.4.6.7.13.9 10.12.11.2.5.1.813 2 3.5.6.7.13.8 10.9.4.12.1.2.1113 2 3.5.6.7.9.12 10.11.4.8.1.2.1313 2 3.5.6.7.11.10 12.9.2.8.1.4.1313 2 3.5.7.8.9.10 12.11.1.2.4.6.1313 3 8.9.10.11.13.12 5.2.3.7.1.6.414 1 1.2.3.8.5.9 13.12.11.6.7.4.1014 1 1.2.3.6.7.9 13.12.11.8.5.4.1014 1 1.2.4.5.6.10 13.12.11.7.9.3.814 1 1.2.4.5.7.9 13.12.10.8.6.3.1114 1 1.2.4.6.7.8 13.12.11.9.3.5.1014 1 2.3.4.5.6.8 12.11.13.10.7.1.914 2 2.5.13.10.8.11 12.9.4.3.6.1.714 2 2.6.9.11.8.13 12.10.1.4.3.5.714 2 2.7.8.9.13.10 12.11.3.5.4.1.614 2 3.4.10.12.7.13 11.9.8.2.1.6.514 2 3.8.4.9.13.12 11.5.6.7.2.1.1014 2 3.5.6.10.12.13 11.8.7.4.2.1.9

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14 2 3.5.7.9.13.12 11.8.6.4.2.1.1014 2 3.5.8.10.12.11 13.6.9.2.4.1.714 2 3.6.7.11.10.12 13.9.1.4.5.2.814 2 3.6.12.9.11.8 13.5.2.1.4.7.1014 2 3.7.8.9.10.12 13.4.11.5.1.2.614 2 4.5.6.10.11.13 12.9.3.8.1.2.714 2 4.5.7.8.12.13 10.11.6.9.2.1.314 2 4.5.10.9.13.8 12.11.3.7.2.1.614 2 4.5.8.9.11.12 10.7.13.3.2.1.614 2 4.6.7.8.11.13 10.12.3.9.1.2.514 2 4.6.7.10.13.9 12.8.1.5.3.2.1114 2 5.6.8.7.13.10 11.9.12.3.2.1.414 2 5.6.7.9.10.12 11.13.1.2.3.4.814 2 5.6.8.10.11.9 13.3.12.2.1.4.715 1 2.3.5.7.10.8 13.11.9.6.4.1.1215 1 2.3.6.7.8.9 13.11.10.4.5.1.1215 1 2.4.5.6.7.11 13.12.8.9.3.1.1015 1 2.4.5.7.8.9 13.11.10.6.1.3.1215 1 3.4.5.6.7.10 13.12.8.9.2.1.1115 1 3.4.5.6.8.9 13.10.11.7.1.2.1215 2 4.8.10.9.13.12 11.5.3.6.1.2.715 2 5.7.8.12.11.13 10.9.2.3.4.1.615 2 5.8.10.9.11.13 12.2.3.4.7.1.615 2 6.8.9.11.10.12 13.1.4.7.2.3.516 1 3.7.9.10.1.12 13.8.4.2.11.6.516 1 3.4.6.10.7.12 13.11.9.1.5.2.816 1 3.4.6.8.9.12 13.11.10.7.2.1.516 1 3.4.7.8.9.11 13.12.6.5.1.2.1016 1 3.5.8.6.11.9 13.12.4.7.1.2.1016 1 4.5.6.7.12.8 13.9.10.3.1.2.1116 1 4.5.6.9.7.11 12.13.8.3.1.2.1016 1 4.5.6.8.9.10 12.13.7.3.2.1.1117 1 5.8.3.11.9.13 12.6.10.4.2.1.717 1 5.6.7.11.8.12 13.9.4.1.2.3.10

APPENDIX C (Computer Program to find all possible VMTL and (a,d)-VATL for C7)

//CYCLE GRAPH with n=7. Vertex-magic and Vertex-antimagic//labelings are found.

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#include<iostream>#include<vector>#include<string>#include<algorithm>#include<stdlib.h>#include<time.h>using namespace std;void saytoyaz(int ii[]);void saytoyazz(int tt[]);void calc(char cd[],char sabitler[]);

int i1,i2,i3,i4,i5,i6,i7,ii[8];int tt[8];int mm=0;int aaa;char cc[7];char sabitler[7];bool onetime=true;bool fff=false;double zz=0,yy=0;

int main(){clock_t start, end;start = clock();for (i1=1;i1<9;i1++){ for (i2=i1+1;i2<10;i2++) { for(i3=i2+1;i3<11;i3++) { for (i4=i3+1;i4<12;i4++) { for (i5=i4+1;i5<13;i5++) { for(i6=i5+1;i6<14;i6++) { for (i7=i6+1;i7<15;i7++) { zz++; ii[1] = i1; ii[2] = i2; ii[3] = i3; ii[4] = i4; ii[5] = i5; ii[6] = i6;

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ii[7]= i7; int gg; gg=1;

for(int s=1;s<15;s++) { if ((i1!= s) && (i2!= s) && (i3!= s) && (i4!= s) && (i5!= s) && (i6!= s) && (i7!= s)) { tt[gg] = s; gg++; } } saytoyaz(ii); saytoyazz(tt); calc(sabitler,cc); int fd; fff=false; for(int i=0;i<5040-1;++i) { for(int fd=0;fd<5040-1;++fd) {

calc(sabitler,cc); next_permutation(sabitler,sabitler+7); if (fff) { fd=6000; i=6000; } } next_permutation(cc,cc+7); }

} } } } } } } cout<<"Number of calculations: "<<yy<<endl; cout<<"Number of solutions: "<<mm<<endl; cout<<"Number of labelings: "<<zz<<endl; end = clock(); cout<<"The time taken was: "<<((end - start) / CLK_TCK)<<endl; cin>>aaa; return 0;}

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void calc(char sabitler[],char cc[]){int zz[7],zx[7],magic[8];for (int a=0;a<7;a++) { if (cc[a] == 'a') { zz[a+1] = 1; } else if (cc[a] == 'b') { zz[a+1] = 2; } else if (cc[a] == 'c') { zz[a+1] = 3; } else if (cc[a] == 'd') { zz[a+1] = 4; } else if (cc[a] == 'e') { zz[a+1] = 5; } else if (cc[a] == 'f') { zz[a+1] = 6; } else if (cc[a] == 'g') { zz[a+1] = 7; } else if (cc[a] == 'h') { zz[a+1] = 8; } else if (cc[a] == 'i') { zz[a+1] = 9; } else if (cc[a] == 'j') { zz[a+1] = 10; }

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else if (cc[a] == 'k') { zz[a+1] = 11; } else if (cc[a] == 'l') { zz[a+1] = 12; } else if (cc[a] == 'm') { zz[a+1] = 13; } else if (cc[a] == 'n') { zz[a+1] = 14; }

if (sabitler[a] == 'a') { zx[a+1] = 1; } else if (sabitler[a] == 'b') { zx[a+1] = 2; } else if (sabitler[a] == 'c') { zx[a+1] = 3; } else if (sabitler[a] == 'd') { zx[a+1] = 4; } else if (sabitler[a] == 'e') { zx[a+1] = 5; } else if (sabitler[a] == 'f') { zx[a+1] = 6; } else if (sabitler[a] == 'g') { zx[a+1] = 7; } else if (sabitler[a] == 'h')

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{ zx[a+1] = 8; } else if (sabitler[a] == 'i') { zx[a+1] = 9; } else if (sabitler[a] == 'j') { zx[a+1] = 10; } else if (sabitler[a] == 'k') { zx[a+1] = 11; } else if (sabitler[a] == 'l') { zx[a+1] = 12; } else if (sabitler[a] == 'm') { zx[a+1] = 13; } else if (sabitler[a] == 'n') { zx[a+1] = 14; } } yy++; magic[1] = zz[7] + zx[1] + zz[1]; magic[2] = zz[1] + zx[2] + zz[2]; magic[3] = zz[2] + zx[3] + zz[3]; magic[4] = zz[3] + zx[4] + zz[4]; magic[5] = zz[4] + zx[5] + zz[5]; magic[6] = zz[5] + zx[6] + zz[6]; magic[7] = zz[6] + zx[7] + zz[7];

int start; int art;

//Check for vertex magic

if ((magic[1]==magic[2])&&(magic[1]==magic[3])&&(magic[1]==magic[4])&&(magic[1]==magic[5])&&(magic[1]==magic[6])&&(magic[1]==magic[7]))

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{ cout<<zz[1]<<"."<<zz[2]<<"."<<zz[3]<<"."<<zz[4]<<"."<<zz[5]<<"."<<zz[6]<<"."<<zz[7]<<"─"<<zx[1]<<"."<<zx[2]<<"."<<zx[3]<<"."<<zx[4]<<"."<<zx[5]<<"."<<zx[6]<<"."<<zx[7]<<" k = "<<magic[1]<<endl; mm = mm+1; fff=true; }

//Check for vertex antimagic

for (start=6;start<29;start++){ for(art=1;art<6;art++) { if (((magic[1] == start) || (magic[2] == start) || (magic[3] == start) || (magic[4] == start) || (magic[5] == start) || (magic[6] ==start) || (magic[7] ==start)) && ((magic[1] == start+art) || (magic[2] == start+art) || (magic[3] == start+art) || (magic[4] == start+art) || (magic[5] == start+art) || (magic[6] ==start+art) || (magic[7] ==start+art)) && ((magic[1] == start+(2*art)) || (magic[2] == start+(2*art)) || (magic[3] == start+(2*art)) || (magic[4] == start+(2*art)) || (magic[5] == start+(2*art)) || (magic[6] ==start+(2*art)) || (magic[7] ==start+(2*art))) && ((magic[1] == start+(3*art)) || (magic[2] == start+(3*art)) || (magic[3] == start+(3*art)) || (magic[4] == start+(3*art)) || (magic[5] == start+(3*art)) || (magic[6] ==start+(3*art)) || (magic[7] ==start+(3*art))) && ((magic[1] == start+(4*art)) || (magic[2] == start+(4*art)) || (magic[3] == start+(4*art)) || (magic[4] == start+(4*art)) || (magic[5] == start+(4*art)) || (magic[6] ==start+(4*art)) || (magic[7] ==start+(4*art))) && ((magic[1] == start+(5*art)) || (magic[2] == start+(5*art)) || (magic[3] == start+(5*art)) || (magic[4] == start+(5*art)) || (magic[5] == start+(5*art)) || (magic[6] ==start+(5*art)) || (magic[7] ==start+(5*art))) && ((magic[1] == start+(6*art)) || (magic[2] == start+(6*art)) || (magic[3] == start+(6*art)) || (magic[4] == start+(6*art)) || (magic[5] == start+(6*art)) || (magic[6] ==start+(6*art)) || (magic[7] ==start+(6*art)))) { cout<<zz[1]<<"."<<zz[2]<<"."<<zz[3]<<"."<<zz[4]<<"."<<zz[5]<<"."<<zz[6]<<"."<<zz[7]<<"−"<<zx[1]<<"."<<zx[2]<<"."<<zx[3]<<"."<<zx[4]<<"."<<zx[5]<<"."<<zx[6]<<"."<<zx[7] <<" start "<<start<<" inc "<<art<<endl; mm = mm+1; fff=true; } }}

}

void saytoyaz(int ii[])

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{for (int a=1;a<8;a++) { if (ii[a] == 1) { cc[a-1] = 'a'; } else if (ii[a] == 2) { cc[a-1] = 'b'; } else if (ii[a] == 3) { cc[a-1] = 'c'; } else if (ii[a] == 4) { cc[a-1] = 'd'; } else if (ii[a] == 5) { cc[a-1] = 'e'; } else if (ii[a] == 6) { cc[a-1] = 'f'; } else if (ii[a] == 7) { cc[a-1] = 'g'; } else if (ii[a] == 8) { cc[a-1] = 'h'; } else if (ii[a] == 9) { cc[a-1] = 'i'; } else if (ii[a] == 10) { cc[a-1] = 'j'; } else if (ii[a] == 11) { cc[a-1] = 'k';

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} else if (ii[a] == 12) { cc[a-1] = 'l'; } else if (ii[a] == 13) { cc[a-1] = 'm'; } else if (ii[a] == 14) { cc[a-1] = 'n'; } }}void saytoyazz(int tt[]){for (int a=1;a<8;a++) { if (tt[a] == 1) { sabitler[a-1] = 'a'; } else if (tt[a] == 2) { sabitler[a-1] = 'b'; } else if (tt[a] == 3) { sabitler[a-1] = 'c'; } else if (tt[a] == 4) { sabitler[a-1] = 'd'; } else if (tt[a] == 5) { sabitler[a-1] = 'e'; } else if (tt[a] == 6) { sabitler[a-1] = 'f'; } else if (tt[a] == 7) { sabitler[a-1] = 'g';

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} else if (tt[a] == 8) { sabitler[a-1] = 'h'; } else if (tt[a] == 9) { sabitler[a-1] = 'i'; } else if (tt[a] == 10) { sabitler[a-1] = 'j'; } else if (tt[a] == 11) { sabitler[a-1] = 'k'; } else if (tt[a] == 12) { sabitler[a-1] = 'l'; } else if (tt[a] == 13) { sabitler[a-1] = 'm'; } else if (tt[a] == 14) { sabitler[a-1] = 'n'; } }}

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