Group Distance Magic Labeling for theCartesian Product of Cycles

byStephen Balamut

MS Candidate: Applied and Computational MathematicsAdvisor: Dalibor Froncek

Department of Mathematics and StatisticsUniversity of Minnesota Duluth

Abstract

Over the past half century, graph labeling has been an emerging area of graph theory. There arestill many open problems. To this end, I have worked toward some new results related to distancemagic labeling. In this paper, I first outline a number of graph labelings with some interestingresults. Secondly, I will introduce two new results on distance magic labeling in a group. Of thetwo results, I demonstrate a distance magic labeling for a special class of graphs when the sums arecalculated using modular arithmetic, and I show that a related case is not group distance magic.

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Contents

1 INTRODUCTION 11.1 Magic Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Graph Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Graph Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 RELATED RESULTS 42.1 Magic Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Edge-Magic Total Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Super Edge-Magic Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Vertex-Magic Total Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Super Vertex-Magic Total Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Totally Magic Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.7 Anti-Magic Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.8 Vertex-Antimagic Total Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.9 Distance Magic Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 MAIN RESULTS 11

4 CONCLUSION 18

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List of Figures

1 Magic Square (n = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Magic Square (n = 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Examples of the Three Fundamental Graph Products [18] . . . . . . . . . . . . . . . . . 34 Small Cases of Distance Magic [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 C4�C4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 A C4�C4 grid representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A different view of C4�C4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 C8�C4 with Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 C4�C4 Demonstrating Neighborhood Sums . . . . . . . . . . . . . . . . . . . . . . . . . 1310 C8�C4 Example of Labeling Main Superdiagonal . . . . . . . . . . . . . . . . . . . . . . 1411 C8�C4 Example of Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1412 C7�C5 Demonstrating Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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1 INTRODUCTION

1.1 Magic Square

Much of what is described in this paper is linked to the 3,000 year old concept of the magic square.This is a square n by n grid of the integers 1 to n2 where each row, each column, and both diagonalssum to the same constant. The ideas here have been linked to the concepts in a number of labelings,and, along with the spirit of the puzzle, the ideas used here inspire expansions of the problem. Forexample, in order to find the sum of a row or column, one can use the summation formula for the first

n2 numbers and divide by n. That is, n2(n2−1)2n =n(n2−1)

2 is the magic constant. Several methods havebeen devised to number magic squares of increasingly large size including using smaller magic squaresto make larger ones. Below are some small examples [26].

6 1 8

7 5 3

2 9 4

Figure 1: Magic Square (n = 3)

9 6 3 16

4 15 10 5

14 1 8 11

7 12 13 2

Figure 2: Magic Square (n = 4)

Transitioning, let us look at this notion from a graph theory point of view. The following sectionincludes some background on graph theory and descriptions of several related concepts that will even-tually lead to some correlated results.

1.2 Basic Definitions

I will be following the notation and terminology in the book A First Look at Graph Theory by Clarkand Holton [5]. Any topics not covered in the book will have notation based on the specific paper citedor a previously cited paper.

Definition 1.1 (Clark, Holton [5]). A graph G=(E(G), V (G)) consists of two finite sets.The vertex set of the graph V (G) is a non-empty set of elements called vertices.The edge set of the graph E(G) is a set of the elements called edges and is possibly empty. Each edgee in E is assigned an unordered pair of vertices (u, v), called the end vertices of e [5].

Definition 1.2 (Clark, Holton [5]). The vertices (sometimes called nodes) are said to be joined byedges. A vertex joined to an edge is said to be incident to the edge.

The way that edges connect vertices in a graph is unrestricted.

Definition 1.3 (Clark, Holton [5]). When an edge starts and ends at the same vertex, the edge iscalled a loop. If two edges have the same starting and ending points, the two edges are called parallel.Graphs with neither loops nor parallel edges are called simple graphs.

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In graph theory, simple graphs are studied most often. In this project we will only deal with simplegraphs. From now on, unless otherwise specified, it is assumed that a graph is simple [5].

Definition 1.4 (Clark, Holton [5]). An edge e of a graph G is said to be incident with the vertex vif v is an end vertex of e. Two edges e and f incident with a common vertex are said to be adjacent.Similarly, two vertices u and v incident with the same edge are said to be adjacent or neighbors.

Definition 1.5 (Clark, Holton [5]). The set of vertices that are adjacent to a vertex v is known as theneighborhood set of vertex v and is denoted N(v).

Definition 1.6 (Clark, Holton [5]). The degree d(v) of a vertex v in a graph G is the number of edgesincident with v. That is, it is the number of times v is an end vertex of an edge. A vertex is either oddor even based on the degree of the vertex.

Definition 1.7 (Clark, Holton [5]). For some positive integer k, a graph G with d(v) = k for each vin G is called a k-regular graph. A regular graph is k-regular for some k.

Definition 1.8 (Clark, Holton [5]). A walk is a sequence of alternating vertices and edges that areincident with the element directly before and after. A walk begins and ends at vertices and passesthrough a sequence of adjacent vertices on the way. If a walk starts and ends at the same vertex, itis called closed; otherwise it is open. If the edges of a walk are distinct, then it is called a trail. Ifthe vertices of a walk are distinct, then it is called a path and is denoted Pn where n is the number ofvertices in the path.

Definition 1.9 (Clark, Holton [5]). A vertex v is said to be connected to a vertex u in a graph G ifthere is a path from v to u. A graph is said to be connected if each pair of vertices in G are connected.A graph that is not connected is called disconnected.

Definition 1.10 (Clark, Holton [5]). A complete graph Kn is a graph in which each vertex is adjacentto every other vertex in the graph.

Definition 1.11 (Clark, Holton [5]). A nontrivial closed trail in a graph G is called a cycle Cn ifits origin and internal vertices are all distinct where n is a positive integer denoting the number ofvertices.

Definition 1.12 (Clark, Holton [6]). A tree is a connected graph that contains no cycles.

A cycle is a 2-regular graph. Later, we will define a graph using the product of cycles that will be4-regular. Mentioned first, though, will be a graph called a hypercube (or k-cube).

Definition 1.13 (Clark, Holton [5]). A hypercube (or k-cube) is a graph whose vertices are orderedk-tuples of 1’s and 0’s such that two vertices are adjacent when the two k-tuples differ in precisely oneposition.

Some of the related results presented below will reference a type of graph called bipartite. This typeof graph can be used to match elements of one group to elements of another.

Definition 1.14 (Clark, Holton [5]). Let G be a graph. Then G is bipartite if the vertex set V of Gcan be partitioned into two non-empty subsets X and Y in such a way that each edge has one end inX and one end in Y . A complete bipartite graph Km,n where m and n are the numbers of vertices inX and Y is a bipartite graph where every vertex in X is adjacent to every vertex in Y .

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1.3 Graph Products

There are three fundamental graph products: Cartesian, direct, and strong. The graphs G and H aresaid to be the factors of the product graph. The Cartesian product will be the focus of this paper andwill be represented with the “�” symbol. The two additional products of graphs are also studied ingraph theory frequently and are also defined below [18].

Definition 1.15 (Hammack, Imrich, Klavzar [18]). Define the Cartesian product G�H to be a graphwith vertex set V (G)×V (H) and (g,h) is adjacent to (g′,h′) if and only if(i) g=g′ and hh′ ∈ E(H) or(ii) h=h′ and gg′ ∈ E(G).

Definition 1.16 (Hammack, Imrich, Klavzar [18]). The direct product of graphs can be defined asG×H and with the vertex set V (G)∪V (H) such that the vertices (g,h) and (g′,h′) are adjacent exactlywhen gg′∈E(G) and hh′∈E(H).

Definition 1.17 (Hammack, Imrich, Klavzar [18]). The strong product of graphs can be defined asG×H such that V (G�H)={(g, h) | g∈V (G) and h∈V (H)} and E(G�H)=E(G�H)∪E(G×H).

Figure 3 shows an example of each product. The graphs of the factors are included to show how eachis constructed.

Figure 3: Examples of the Three Fundamental Graph Products [18]

1.4 Graph Labelings

Graph labelings are ways to assign integers to vertices and/or edges of a graph under some rule orrestriction. This area of mathematics has only been around for about a half century and has alreadybeen published in more than 1,000 papers. A collection of these results has been gathered by JosephGallian in A Dynamic Survey of Graph Labeling [12].

Definition 1.18. A graph labeling is an assignment f of integers to vertices and/or edges of a graph.

Research on labelings tends to focus on classes of graphs that can be generalized in some way. Simplegraphs (graphs with no parallel edges or loops), regular graphs (graphs with the same degree at eachvertex), symmetric graphs, and vertex-transitive graphs (graphs where the vertices are indistinguish-able from those around it) are all examples of this. For the results in this paper, graphs that areproducts of cycles will be the focus.

Before looking at a number of magic type labelings, I will first introduce some of the Rosa typesof graph labelings, which were among the first to be defined.

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Definition 1.19. Let G be a graph with n edges. Then a ρ-labeling of G is a one-to-one functionf : V (G) → {0, 1, ..., 2n} such that {min{|f(u) − f(v)|, 2n+1−|f(u) − f(v)|} : {u, v} ∈ E(G)}={1, 2, ..., n} [8].

As we will see with graceful labelings, the difference of vertex labels is used to determine the edgelabel, but this time the labels could be the difference subtracted from 2n− 1.

This next labeling and its more restrictive variations described below were introduced by Rosa in1967. This labeling was originally called β-valuation by Rosa. Now it is referred to as graceful la-beling. Assigning integers to each vertex and each edge, this type of labeling has assigned values foredges based on the value of their incident vertices.

Definition 1.20. A graceful labeling is a ρ-labeling except the vertex labels are in {0, 1, 2, ..., n} andthe value of each edge, uv, is |f(u)− f(v)| [13].

In other words, in order for a graph to be considered graceful, there still needs to be an injection fromthe first n integers to the edges where n is the number of edges. Most labelings have their origins inthis type [13].

An interesting special case of the graceful labeling is the α-labeling.

Definition 1.21. An α-labeling is a graceful labeling with the additional condition that there existssome constant k such that the following is true for each edge xy: f(x) ≤ k < f(y) or f(y) ≤ k < f(x).

This is a way to balance a graph labeling and is unsurprisingly sometimes called balanced or strongly-graceful [14].

Finally, a harmonious labeling is an additive analogy of a graceful labeling. It is identical to gracefullabeling except edge labels are computed f(x) + f(y) (mod n) where n is the number of edges. Thus,to calculate the label of a particular edge, simply compute the sum of the incident vertices modulo n[12].

Next, I will describe some related work done with graph labeling. At the end of the section I willintroduce the topic and paper that inspired my new results.

2 RELATED RESULTS

There are numerous types of magic labelings in graph theory. Each has its own conditions based onthe labels at each vertex and/or edge. The following are some different types of magic labelings thathave been studied along with some interesting results for each.

2.1 Magic Labeling

In the 1960’s magic labelings began being studied. Naturally, a number of variations were created. Agraph is said to be semi-magic if the edges can be labeled in a way that the sum of the incident edgesis the same for every vertex chosen. A semi-magic graph becomes a magic graph when the edges arelabeled with distinct positive integers. As might be expected, a supermagic graph is a magic graph inwhich the edges are consecutive positive integers [15]. Ivanco showed the following.

Theorem 2.1 (Ivanco [20]). C2n�C2k, n, k ≥ 2 is supermagic.

Theorem 2.2 (Ivanco [20]). Cn�Cn for each integer n ≥ 3 is supermagic.

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Theorem 2.3 (Ivanco [20]). Qn is supermagic if either n=1 or n ≥ 4 and n ≡ 0 (mod 2).

He also conjectured the following result that deals with the Cartesian products of cycles.

Conjecture 2.4 (Ivanco [20]). Cn�Ck is supermagic.

2.2 Edge-Magic Total Labeling

An edge-magic labeling is called edge-magic total (abbreviated EMT) to distinguish from other magictype labelings. It is a bijection f from the set V ∪E to the set of integers, {1,2,· · · ,|V |+|E|} such thatfor every edge uv, f(u) + f(v) + f(uv) = h where h is a constant. We mentioned that for a 2-regulargraph, an EMT labeling is the same as a vertex magic total labeling as described below [15].

When studying graph labelings, mathematicians often work with certain types or classes of graphs. Inthe spirit of this, Ngurah, Baskoro, Simanjuntak, and Uttunggadewa studied kC4-snakes.

Definition 2.1 (Ngurah, Baskoro, Simanjuntak, Uttunggadewa [24]). A kC4-snake is a connectedgraph with k blocks where the blocks are repeated components of the graph. Each of the blocks isisomorphic to the cycle C4 such that the block-cut-vertex graph is a path (a graph whose verticescorrespond to the the blocks and are adjacent if the blocks were adjacent). One could think of this asa row of diamonds.

Theorem 2.5 (Ngurah, Baskoro, Simanjuntak, Uttunggadewa [24]). kC4-snakes are EMT.

Kotzig and Rosa also showed some interesting results.

Theorem 2.6 (Kotzig, Rosa [15]). Complete bipartite graphs Km,n are EMT for all m,n.

Theorem 2.7 (Kotzig, Rosa [15]). Cycles Cn are EMT for n ≥ 3.

Wallis, Baskoro, Miller, and Slamin showed some other interesting results.

Theorem 2.8 (Wallis, Baskoro, Miller, Slamin [27]). Odd cycles Cn have EMT labelings for thefollowing magic constants:

1. 12(5n+ 3)

2. 12(7n+ 3)

3. 3n+ 1

4. 3n+ 2

Theorem 2.9 (Wallis, Baskoro, Miller, Slamin [27]). Even cycles Cn have EMT labelings for the magicconstant 1

2(5n+ 4).

Theorem 2.10 (Wallis, Baskoro, Miller, Slamin [27]). Cycles Cn of length divisible by 4 have EMTlabelings for the following magic constants:

1. 12(7n+ 2)

2. 3n

3. 3n+ 3

They also proved a result about kites.

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Definition 2.2 (Wallis, Baskoro, Miller, Slamin [27]). A kite is a cycle with a path connected to oneof its vertices. A kite of tail length 1 is a cycle with a path of length 1 connected to one of the verticesin the cycle.

Theorem 2.11 (Wallis, Baskoro, Miller, Slamin [27]). Kites with tail length 1 are EMT.

Perhaps the most interesting conjecture about EMT labelings is the following. Although no one hasbeen able to prove or disprove it, the conjecture continues to be a popular unsolved problem.

Conjecture 2.12 (Baca, Miller [2]). Every tree is EMT.

2.3 Super Edge-Magic Labeling

A super edge-magic labeling (SEM) is the same as the edge-magic total labeling with the additionalcondition that the vertices map to the smallest integers {1, 2,. . ., |V |}. Below, super vertex-magic totallabelings will have the same additional condition.

Enomoto, Llado, Nakamigawa, and Ringel showed the following about SEM labeling.

Theorem 2.13 (Enomoto, Llado, Nakamigawa, Ringel [9]). Cycles Cn are SEM if and only if n isodd.

Definition 2.3 (Clark, Holton [5]). A wheel graph Wn is a graph made of a cycle Cn−1 and a singlevertex where there is a path of length 1 from each vertex in the cycle to the single vertex.

I will not that other authors will define a wheel graph using the cycle Cn rather than Cn−1.

Theorem 2.14 (Enomoto, Llado, Nakamigawa, Ringel [9]). There does not exist an n such that thewheel graph Wn is super edge-magic.

2.4 Vertex-Magic Total Labeling

A vertex-magic total labeling (abbreviated VMT) is a bijection from the set V ∪E to {1, 2,. . ., |V |+|E|}.For this labeling there is some constant weight h such that for each vertex x the sum of the label forx and the labels for all the edges incident to x is h [2].

Related to the results later in this paper, Froncek, Kovar, and Kovarova showed the following aboutthe Cartesian product of cycles. Later it will be shown that this result does not necessarily hold forsimilar magic type labelings; specifically, products of two odd cycles are not distance magic.

Theorem 2.15 (Froncek, Kovar, Kovarova [11]). For each m,n ≥ 3 and at least one of m,n odd,there exists a VMT labeling of Cm�Cn for each of the following magic constants:

1. 12(15mn+m+ 4)

2. 12(17mn+ 5)

One open problem in this area comes from the paper Vertex-Magic Labelings of Regular Graphs byMacDougal [22]. Since the conjecture was made, a number of constructions for regular graphs havebeen formulated, but it has never been proven. Conversely, no counterexample has been found either.

Conjecture 2.16 (Baca, Miller [2]). Every regular graph has a VMT labeling besides K2 and 2K3

Petr Kovar has since put forward a number of results working toward this conjecture. Although hedoes not cover every case, he was able to show that many regular graphs are VMT. The followingtheorems demonstrate this.

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Theorem 2.17 (Kovar [21]). Let G be a (2 + s)-regular graph such that it contains an s-regular factorG′ which allows a VMT labeling with magic constant h and vertex labels being consecutive integersstarting at k. Then G is VMT.

Theorem 2.18 (Kovar [21]). Let G be a (2r+s)-regular graph such that it contains an s-regular factorG′ which allows a VMT labeling with magic constant h′ and vertex labels being consecutive integersstarting at k. Then G is VMT.

2.5 Super Vertex-Magic Total Labeling

Similar to the vertex-magic total labeling, a super vertex-magic total labeling is one with the additionalcondition that the smallest integers are the vertex labels and the largest are the edge labels. That is,the vertices are labeled using the set {1, 2,. . ., |V |}. Some interesting results are given in a paper byMacDougal, Miller, and Sugeng [22]; however, as is expected they are more restricted under the newcondition.

Theorem 2.19 (MacDougal, Miller, Sugeng [22]). A cycle Cn has a super vertex magic total labelingif and only if n is odd.

They also showed that a number of different graphs do not have a super VMTL such as completebipartite graphs and graphs with a vertex of degree one (including trees) [22]. Below, we will see anegative result for distance magic labelings.

2.6 Totally Magic Labeling

A totally magic graph is one in which it has a vertex-magic total labeling and an edge-magic totallabeling. Remembering that each vertex is dependent on the incident edges and each edge is dependenton its incident labels, this is obviously a difficult labeling. Many results are small special cases so far.

Exoo, Ling, McSorley, Phillips, and Wallis studied how cycles and vertices of degree one within agraph can affect the way the graph must be labeled.

Theorem 2.20 (Exoo, Ling, McSorley, Phillips, and Wallis [10]). The only totally magic cycle is C3

(or K3).

Theorem 2.21 (Exoo, Ling, McSorley, Phillips, and Wallis [10]). The only connected totally magicgraph containing a vertex of degree one is P3. Consequently, the only totally magic trees are P3 andK1.

Theorem 2.22 (Exoo, Ling, McSorley, Phillips, and Wallis [10]). If a totally magic graph contains atriangle (C3), then the sum of the edge labels incident to any one vertex of the triangle and outside thetriangle is the same no matter which vertex is chosen.

Now, to take a look at a type of labeling that assigns labels to the vertices similarly to magic labeling,I have included anti-magic labeling. In this case, the concentration shifts from equality to pairwisedistinct.

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2.7 Anti-Magic Labeling

An anti-magic labeling of a graph is essentially the opposite of a magic labeling. That is, it is a bijec-tion from the vertices to consecutive integers, but the sums found must be distinct. Each vertex summust be different from every other vertex sum [28].

Wang and Hsiao showed a number of results including the Cartesian products of basic graphs. Aswe saw with some other magic type labelings, this type of product will continue to be part of thetheme of this paper.

Theorem 2.23 (Wang Hsiao [28]). Pm�Pn (lattice grid graphs) are antimagic for m ≥ n ≥ 2.

Expanding, they also showed a result including a cycle as one of the factors.

Theorem 2.24 (Wang Hsiao [28]). Cm�Pn (prism grids) are antimagic.

Theorem 2.25 (Wang Hsiao [28]). Other antimagic graphs include

1. Paths Pn for n ≥ 3

2. Cycles Cn for n ≥ 3

3. Hypercube graphs Qn

A strong result in this area was shown by Cheng.

Theorem 2.26 (Cheng [4]). All Cartesian products of two or more regular graphs of positive degreeare antimagic.

This result about antimagic graphs is very interesting since nothing like this has been shown for distancemagic graphs. After describing a number of graphs that are antimagic, Hartsfield and Ringel had thefollowing conjecture.

Conjecture 2.27 (Hartsfield, Ringel [19]). Every connected graph, excluding K2, is antimagic.

A number of researchers have unsuccessfully tried to show this to be false. At the same time, given aconnected graph, it is typically not difficult to find an antimagic labeling for it. Moreover, there areoften multiple such labelings. With that in mind, as with magic labelings, more restrictions have beenadded to form new types of anti-magic graphs [2].

2.8 Vertex-Antimagic Total Labeling

A vertex-antimagic total labeling is a bijection from the set V ∪ E to the set of integers, {1, 2,. . .,|V |+|E|} where the weights of all the vertices are distinct with the additional restriction that all theweights form an arithmetic progression. Interestingly, to show a relation to magic labelings, we havethe following result.

Theorem 2.28 (Baca, Miller [2]). Every super-magic graph has a vertex-antimagic total labeling.

To move back to the perspective of magic labeling, next I define the type of labeling that is the focalpoint of this paper. Distance magic labeling is the natural way to extend magic type labeling to thevertex set only.

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2.9 Distance Magic Labeling

Also known as sigma labeled or 1-vertex magic vertex labeled, a distance magic labeled graph is saidto be distance magic. The labeling f is said to have a magic constant w. A graph G of order n has adistance magic labeling if there is a bijection f : V → {1, 2,. . ., n } there is a positive integer w suchthat

∑y∈N(x) f(y) = w for every x ∈ V where N(x) is the neighborhood set of vertex x [1].

It was shown by Miller, Rodger, and Simanjuntak that some common types of graphs are rarelydistance magic. They proved the following.

Theorem 2.29 (Miller, Rodger, Simanjuntak [23]). Some results.

1. Pn is distance magic if and only if n ∈ {1, 3}.

2. Cn is distance magic if and only if n = 4.

3. Kn is distance magic if and only if n = 1.

4. Wn is distance magic if and only if n = 4.

Figure 4: Small Cases of Distance Magic [23]

Much of the work in my project is inspired by the paper by S.B. Rao, T. Singh, and V. Parameswaran.They proved this as the main result of the paper.

Theorem 2.30 (Rao, Singh, and Parameswaran [25]). The graph Cn�Ck, n, k ≥ 3 is distance magiciff n = k and k ≡ 2 (mod 4).

Of course, this is under the group, the integers. For the main results, I will be working with finitecyclic groups, but I will be using some of the ideas from their proofs to show my results [25].

Next, I will outline the proof given by Rao, Singh, and Parameswaran. They used a series of lemmasin order to eventually prove that Cn�Ck, n, k ≥ 3 is distance magic iff n = k and k ≡ 2 (mod 4). Theideas from the first lemma are used later on to prove the new result. Here are the lemmas they used.

Lemma 1. The graph Cn�Ck, n, k ≥ 3 is not distance magic if n and k are both odd.

Lemma 2. The graph Cn�Ck, n ≤ k where n ≡ 0 (mod 4) is not distance magic.

Lemma 3. Cn�Ck, n, k ≥ 3 where n 6=k and n ≡ 2 (mod 4) is not distance magic.

Lemma 4. There is a general distance magic labeling for Cn�Ck, n, k ≥ 3 where n = k and k ≡ 2(mod 4) which shows sufficiency.

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I note that the first three lemmas of the proof show the necessity of the conditions. The fourth lemmashows the sufficiency. That is, they provided a simple labeling under the conditions to finish the proof.Finally, at the end of their paper, they expanded by adding an additional theorem.

Theorem 2.31 (Rao, Singh, and Parameswaran [25]). Km�Kn, m ≥ 2, n ≥ 3 is not distance magic.

As a remark, we can note that K2�K2 is distance magic and is shown above as C4 [25].

Looking at some other interesting results, Beena proved the following two theorems.

Theorem 2.32 (Beena [3]). Given two positive integers m and n such that m ≤ n, the completebipartite graph Km,n is distance magic if and only if the following are true.

1. m+ n ≡ 0 or 3 (mod 4) and

2. either n ≤ b(1 +√

2m− 12c or 2(2m+ 2n+ 1)2 = 1.

Theorem 2.33 (Beena [3]). The product of paths Pn�Pk is not distance magic for all n, k ≥ 3.

Miller, Rodger, and Simanjuntak included in their paper some interesting negative results for labeling.

Theorem 2.34 (Miller, Rodger, Simanjuntak [23]). If G contains two vertices u and v where |N(u)∩N(v)|= d(v)− 1 = d(u)− 1, then G is not distance magic.

That is, if two vertices share all their neighbors except for one for each of them, then the graph is notdistance magic.

Theorem 2.35 (Miller, Rodger, Simanjuntak [23]). If G has n vertices with a maximum degree of ∆and minimum degree of δ, then G does not have a labeling when ∆(∆ + 1) > δ(2n− δ + 1).

Theorem 2.36 (Miller, Rodger, Simanjuntak [23]). Every k-regular graph with odd k does not have amagic labeling.

These types of results can make it easy for future researchers to rule out many types of graphs as magic.

Now that we have seen a number of different results about graph labeling including the productsof graphs, one might ask if the product of cycles, other than the ones mentioned above, can ever be“distance magic” if we relax the definition somehow. If so, how can we work around the results by Rao,Singh, and Parameswaran? If not, what makes it impossible? I will demonstrate proofs for two newresults that answer these questions, and they will take the form of a construction and a non-existenceresult, respectively.

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3 MAIN RESULTS

In an attempt to expand the work of Rao, Singh, and Parameswaran mentioned above, I introduce twonew results with the perspective of finite groups. That is, consider a new type of labeling based ondistance magic labeling. When the vertex weights are calculated under modular arithmetic, call thislabeling “group distance magic.” The first result of this paper is to define a construction for a groupdistance magic labeling for Cn�Ck, n ≡ 0 (mod 4) and k=4. Secondly, I show that when n and k areboth odd there is not a group distance magic labeling under the group Znk.

Definition 3.1. A graph G of order n has a group distance magic labeling if there is a bijectionf : V → Zn and a positive integer w such that

∑y∈N(x) f(y) = w for every x ∈ V where N(x) is the

neighborhood set of vertex x, w ∈ Zn, and addition is in Zn.

The first theorem will be proved by demonstrating a labeling for the graphs in question. In order todo this I will define a number of concepts and symbols as part of the first proof. Eventually, there willbe a distinct magic constant for this type of graph. In the proof of the second theorem, I will showthat the graphs in question are not group distance magic by demonstrating an algebraic contradiction.This will use a combination of graph theory and algebraic concepts to show that every graph of itstype would have two vertices with the same label.

Before constructing the labeling to prove Theorem 3.1, I will define some main concepts and gen-eral methods for working with products of cycles. I will show how to represent these graphs as agrid, how to differentiate notation for vertices, and how to calculate weights at vertices based on theserepresentations. First, I will look at the grid representation.

Given the torus nature of these graphs, the labelings can be easily represented using an n by k gridwhere boxes represent vertices, and vertices are adjacent if the boxes share a grid line. Since eachvertex is adjacent to exactly four other vertices, consider boxes on the corners and sides to be adjacentto the boxes directly opposite them. The grid (Figure 6) represents the graph (Figure 5) where vertexa is adjacent to vertices b, c, d, and e. That is, the neighborhood set N(a) is made of b, c, d, and e.

Figure 5: C4�C4

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a e # c

d # # #

# # # #

b # # #

Figure 6: A C4�C4 grid representation

Remembering that this graph has a torus structure, a different grid view of the same graph might looklike Figure 7.

# b # #

c a e #

# d # #

# # # #

Figure 7: A different view of C4�C4

Given this new representation for these graphs, denote the vertex labels of the graph as ai,j where j isgiven modulo 4. Figure 8 shows an example.

a0,0 a0,1 a0,2 a0,3

a1,0 a1,1 a1,2 a1,3

a2,0 a2,1 a2,2 a2,3

a3,0 a3,1 a3,2 a3,3

a4,0 a4,1 a4,2 a4,3

a5,0 a5,1 a5,2 a5,3

a6,0 a6,1 a6,2 a6,3

a7,0 a7,1 a7,2 a7,3

Figure 8: C8�C4 with Indexing

Define f(ai,j) to be the label of a vertex ai,j , and define the neighborhood sum of a vertex ai,j by σi,jwhere i is taken modulo n and j is taken modulo k. In this case, k = 4. As an example, Figure 9 showsσ1,0=(0+7+13+6) modulo 16, which equals 26 modulo 16. Finally, 26 is congruent to 10 (mod 16).

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0 1 5 12

σ1,0 6 3 7

13 4 8 9

11 15 10 14

Figure 9: C4�C4

Demonstrating Neighborhood Sums

The method for labeling Cn�Ck graphs is based on the property that neighborhood sums must beequal throughout. For example, σ1,1=σ2,2.

More generally, remembering that the indices are calculated (mod n) and (mod k),

σa,b=σa+1,b+1.

The direct consequence of this, for example, is that

f(a1,0)+f(a0,1)=f(a3,2)+f(a2,3).

More generally, remembering that the indices are calculated (mod n) and (mod k),

f(ac,d)+f(ad,c)=f(ac+2,d+2)+f(ad+2,c+2).

With these general definitions at hand, I can now demonstrate a construction of group distance magiclabeling for this type of graph.

Definition 3.2. Given a Cn�C4 graph, the main superdiagonal is the set of vertices ai,j such thati = j (mod 4). The lth superdiagonal is the set of vertices ai,j such that i = j + l (mod 4).

Theorem 3.1. A Cn�C4 graph where n ≡ 0 (mod 4) has a group distance magic labeling.

Proof. Label the sequence of vertices ai,j where i = j and j is taken modulo 4 by the following method.First, let a0,0 = 0 and a1,1 = 4. Next, for i even, let ai,j = (8 + ai−2,j−2)(mod 4n). Similarly, for iodd, let ai,j = (−8 + ai−2,j−2)(mod 4n). Although the labeling is for a torus, for simplicity, think ofthe vertices ai,j where i = j as the main superdiagonal of the given grid. For k = 8, this part of thelabeling is shown in Figure 10.

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0 # # #

# 4 # #

# # 8 #

# # # 28

16 # # #

# 20 # #

# # 24 #

# # # 12

Figure 10: C8�C4

Example of Labeling Main Superdiagonal

Skipping over a vertex, label the second superdiagonal (a0,2, a1,3, a2,0, ...) based on the already labeledvertices. Vertices ai,j where j = i+ 2 and j is taken modulo 4 are labeled using a common sum. Thatis, ai,j + ai+1,j−1 = 4n − 1. For example, when n = 8, the sum of the vertex (ai,j) and its “partner”(ai+1,j−1) is 31. As a consequence, subtracting the given number (specifically, a label from the mainsuperdiagonal) from 31 will give the desired label on the second superdiagonal.

Labeling the remaining vertices is a matter of simply adding or subtracting 2 from certain neigh-bors. For the first superdiagonal (where j = i+ 1 and j is taken modulo 4), ai,j = ai−1,j − 2. For thethird superdiagonal (where j = i + 3 and j is taken modulo 4), ai,j = ai−1,j + 2. The labeled graphwhere n = 8 is shown in Figure 11.

0 29 27 14

2 4 25 23

3 6 8 21

1 15 10 28

16 13 11 30

18 20 9 7

19 22 24 5

17 31 26 12

Figure 11: C8�C4

Example of Labeling

To show that this labeling is group distance magic for all graphs of this type, consider the sum at anarbitrary vertex in each of the four diagonal cases below where i is 0, 1, 2, or 3. Also, for indexingsimplicity, define m as a non-negative integer less than n

4 . For each of these, when computed withmodular arithmetic, there is a relatively simple formula (based on the labeling method) showing thatall the sums are equal. For each of the four cases, the sum of its neighbors is shown in terms of thelabeling and in reference to the given vertex. Again, 4n− 1 is used to label the second superdiagonalas described above. It is used as a reference for two of the neighbors in each case.

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1. The weight for each vertex on the main superdiagonal of the grid, σ4m+i,i, is the sum of thefollowing:

(a) [4n− 1− f(a4m+i−1,i−1)− 2] (above)

(b) [f(a4m+i,i) + 2] (below)

(c) [f(a4m+i−1,i−1) + 2] (left)

(d) [4n− 1− f(a4m+i,i)− 2] (right)

Adding the four neighbors, we get[f(a4m+i,i) + 2] + [f(a4m+i−1,i−1) + 2] + [4n− 1− f(a4m+i,i)− 2] + [4n− 1− f(a4m+i−1,i−1)− 2]= 8n− 2 (mod 4n) = w

2. The weight for each vertex on the first superdiagonal, σ4m+i,i+1 is the sum of the following:

(a) [f(a4m+i,i+1) + 2] (above)

(b) [4n− 1− f(a4m+i,i+1)− 2] (left)

(c) [4n− 1− f(a4m+i+1,i+1+1)− 2] (below)

(d) [f(a4m+i+1,i+1+1) + 2] (right)

Adding the four neighbors, we get[f(a4m+i,i+1)+2]+[f(a4m+i+1,i+1+1)+2]+[4n−1−f(a4m+i,i+1)−2]+[4n−1−f(a4m+i+1,i+1+1)−2]= 8n− 2 (mod 4n) = w

3. The weight for each vertex on the second superdiagonal, σ4m+i,i+2 is the sum of the following:

(a) [4n− 1− f(a4m+i−1,i+2−1) + 2 + 8] (above)

(b) [f(a4m+i−1,i+2−1)− 2] (left)

(c) [f(a4m+i,i+2)− 2] (below)

(d) [4n− 1− f(a4m+i,i+2) + 2− 8] (right)

Adding the four neighbors, we get[f(a4m+i,i+2)−2]+[f(a4m+i−1,i+1)−2]+[4n−1−f(a4m+i,i+2)+2−8]+[4n−1−f(a4m+i−1,i+1)+2 + 8] = 8n− 2 (mod 4n) = w

4. The weight for each vertex on the third superdiagonal, σ4m+i,i+3 is the sum of the following:

(a) [f(a4m+i,i+3)− 2] (above)

(b) [4n− 1− f(a4m+i,i+3)− 8 + 2] (left)

(c) [4n− 1− f(a4m+i+1,i) + 8 + 2] (below)

(d) [f(a4m+i+1,i)− 2] (right)

Adding the four neighbors, we get[f(a4m+i,i+3)−2]+[f(a4m+i+1,i)−2]+[4n−1−f(a4m+i,i+3)−8+2]+[4n−1−f(a4m+i+1,i)+8+2]= 8n− 2 (mod 4n) = w

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That is, all vertices have a weight of (8n − 2) modulo 4n which is congruent to (4n − 2) modulo 4nunder the given labeling. This means there is a group distance magic labeling for each graph of thistype.

Next, one might wonder what happens when the cycles have an odd number of vertices. Interestingly,changing to this gives a quite different result. Before introducing my second result, I must prove aproposition for later use.

Proposition 3.2. The mapping φ(x) = 2x from Z2n+1 to Z2n+1 is a bijection. As a result, forelements a and b, 2a = 2b implies a = b under the mapping.

Proof. Since a mapping from Z2n+1 to Z2n+1 is a bijection if and only if it is a surjection, I will proveonly the surjective property. Notice that for any element x ∈ Z2n+1, either x = 2z and 0 ≤ z ≤ n orx = 2z + 1 and 0 ≤ z ≤ n− 1.I will show for all x in Z2n+1 there exists a y such that φ(y) = 2y = x.To do this, I will use two cases.For the case when x = 2z and 0 ≤ z ≤ n, y = z is the preimage since φ(y) = 2y = 2z = x.For the case when x = 2z + 1 and 0 ≤ z ≤ n − 1, y = n + z + 1 is the preimage since φ(y) = 2y =2(n+ z + 1) = (2n+ 1) + (2z + 1) = 2z + 1 = x.Thus, φ(x) is a bijection. For the purposes of this paper, 2x = 2y ⇒ x = y.

Now that I have shown this injective property, I can prove my second result. The following is mysecond theorem based on the work by Rao, Singh, and Parameswaran [25].

Theorem 3.3. A Cn�Ck graph where n and k are both odd is not group distance magic.

Proof. I will use the method from the proof of Lemma 1 in [25] to show these graphs are not groupdistance magic.Considering the neighborhood sums of a1,1 and a2,2, as was described above,

f(a1,0)+f(a0,1)=f(a2,3)+f(a3,2).

Using the same logic,

f(a2,3)+f(a3,2)=f(a4,5)+f(a5,4).

These equalities continue such that for some integer x the sum of each pair in the pattern is equal tothe sum

f(a2x+1,2x+0)+f(a2x+0,2x+1)

when calculating the indices under the appropriate finite group. In other words, thinking of the graphas a torus, the pairs of equal sums continue to wrap around. To show there exists an x such that thesum f(a2,1)+f(a1,2) is included, we have the congruences,

2x+ 0 ≡ 1 (mod n)

2x+ 1 ≡ 2 (mod k)

2x+ 1 ≡ 2 (mod n)

2x+ 0 ≡ 1 (mod k)

which reduce to

2x ≡ 1 (mod n)

2x ≡ 1 (mod k).

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This implies there exist positive integers s and t such that 2x=nt + 1=ks + 1 ⇒ nt=ks. Letd=GCD(n, k). It follows that the equation is satisfied by t=k

d and s=nd . Substituting, 2x=nk

d +1. Now,

n, k, and d are all odd integers; therefore, x=12(nkd + 1) exists as an integer to satisfy f(a2,1)+f(a1,2).

Next, by the same logic, it can be shown that the analogous is true for sums along perpendiculardiagonals of the torus graph. That is, repeating the proof starting with the neighborhood sums of a1,1

and a2,0, there exists an x such that f(a0,1)+f(a1,2)=f(a2x+0,2x+1)+f(a2x+1,2x+2)= f(a1,0)+f(a2,1).

To summarize, it has now been shown that

f(a1,0) + f(a0,1) = f(a2,1) + f(a1,2)

and

f(a0,1) + f(a1,2) = f(a1,0) + f(a2,1).

Visually, for C7�C5 the labels on the grid are shown in Figure 12.

# a0,1 # # #

a1,0 # a1,2 # #

# a2,1 # # #

# # # # #

# # # # #

# # # # #

# # # # #

Figure 12: C7�C5

Demonstrating Layout

Remembering that the labels are integers, we can have a system of equations and solve. Subtracting

f(a1,0)+f(a0,1)=f(a2,1)+f(a1,2)f(a0,1)+f(a1,2)=f(a1,0)+f(a2,1)

gives

f(a1,0)− f(a1,2)=f(a1,2)− f(a1,0).

Solving the new equation, we have

2f(a1,0)=2f(a1,2).

Applying Proposition 3.2,

f(a1,0)=f(a1,2).

In words, two vertices must have the same label. This equality contradicts the assumption that thereis a bijection from the labeling to the integers modulo nk. As a result, there is no labeling for Cn�Ck

while both n and k are odd.

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4 CONCLUSION

We have seen that there are many types of graph labelings with highly varied results. Small changescan have a substantial effect on whether or not a graph can be labeled in a certain way. In this case,by introducing a new magic type labeling as a variation on an established one, I was able to label atype of graph that was not possible before the change. At the same time, I showed that the result didnot change for odd cycle graphs when using group distance magic.

Stated again, there exists a group distance magic labeling for Cn�Ck graphs where k = 4 and n ≡ 0(mod 4). Moreover, the magic constant w equals 4n− 2. When n and k are both odd, Cn�Ck is notgroup distance magic.

These results open up new questions for future research. The next logical area to explore wouldbe the case of Cn�Ck where both n and k are equivalent to 0 modulo 4. After that, one would wonderabout the cases where n and k are different combinations of 0 modulo 4 and 2 modulo 4. That is, theideas to explore are the cases not covered by the negative result.

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