heavy cycles passing through some specified vertices in weighted graphs

Heavy Cycles PassingThrough SomeSpecified Verticesin Weighted Graphs

Jun Fujisawa,1 Kiyoshi Yoshimoto,2

and Shenggui Zhang3

1DEPARTMENT OF MATHEMATICS, KEIO UNIVERSITY

HIYOSHI 3-14-1, KOHOKU-KU

YOKOHAMA 223-8522, JAPAN

E-mail: [email protected]

2DEPARTMENT OF MATHEMATICS, COLLEGE OF SCIENCE

AND TECHNOLOGY, NIHON UNIVERSITY

TOKYO 101-8308, JAPAN


3DEPARTMENT OF APPLIED MATHEMATICS

NORTHWESTERN POLYTECHNICAL UNIVERSITY

XI’AN, SHAANXI 710072, P. R. CHINA


Received March 11, 2003; Revised July 30, 2004

Published online in Wiley InterScience(www.interscience.wiley.com).

DOI 10.1002/jgt.20066

Abstract: A weighted graph is one in which every edge e is assigneda nonnegative number, called the weight of e. The sum of the weightsof the edges incident with a vertex v is called the weighted degree of v.

——————————————————

Contract grant sponsor: KAKENHI (to K.Y.); Contract grant number: 14740087.

� 2005 Wiley Periodicals, Inc.

93

The weight of a cycle is defined as the sum of the weights of its edges.In this paper, we prove that: (1) if G is a 2-connected weighted graph suchthat the minimum weighted degree of G is at least d , then for every givenvertices x and y , either G contains a cycle of weight at least 2d passingthrough both of x and y or every heaviest cycle in G is a hamiltonian cycle,and (2) if G is a 2-connected weighted graph such that the weighted degreesum of every pair of nonadjacent vertices is at least s, then for every vertexy , G contains either a cycle of weight at least s passing through y or ahamiltonian cycle. AMS classification: 05C45 05C38 05C35.� 2005 Wiley Periodicals, Inc. J Graph Theory 49: 93–103, 2005

Keywords: weighted graph; (heavy, hamiltonian) cycle; minimum (weighted) degree; (weighted)

degree sum

1. TERMINOLOGY AND NOTATION

A weighted graph is one in which every edge e is assigned a nonnegative number

wðeÞ, called the weight of e. The set of all the neighbors of a vertex v in G is

denoted by NGðvÞ or simply NðvÞ, and its cardinality by dGðvÞ or dðvÞ. The

weighted degree of v is defined by:

dwGðvÞ ¼X

x2NðvÞwðvxÞ:

When no confusion occurs, we denote dwGðvÞ by dwðvÞ. For a subgraph H of G,

VðHÞ and EðHÞ denote the sets of vertices and edges of H, respectively. The

weight of H is defined by:

wðHÞ ¼X

e2EðHÞwðeÞ:

An unweighted graph can be regarded as a weighted graph in which every edge e

is assigned weight wðeÞ ¼ 1. Thus, in an unweighted graph, dwðvÞ ¼ dðvÞ for

every vertex v, and the weight of a cycle is simply the length of the cycle.

If G is a noncomplete graph, we define:

�w2 ðGÞ ¼ minfdwðuÞ þ dwðvÞ j u and v are nonadjacentg:

If G is complete, then �w2 ðGÞ is defined as 1.

An ðx1; x2Þ-path is a path joining vertices x1 and x2. An ðx1; y; x2Þ-path is an

ðx1; x2Þ-path passing through a vertex y. Moreover, when P is an ðx; yÞ-path such

that y 2 M and VðPÞ \ VðMÞ ¼ fyg, we call P an ðx;MÞ-path.

A y-cycle is a cycle passing through y and an ðx; yÞ-cycle is a cycle passing

through both of the vertices x and y. A component containing a vertex y is called a

y-component.

94 JOURNAL OF GRAPH THEORY

For simplicity, we denote jVðLÞj by jLj, and ‘‘ui 2 VðLÞ’’ and ‘‘G� VðLÞ’’ are

written by ‘‘ui 2 L’’ and ‘‘G� L’’ respectively for a subgraph L. The set of

neighborsS

v2L NGðvÞnVðLÞ is written by NGðLÞ or NðLÞ, and for a subgraph

H � G, NGðLÞ \ VðHÞ is denoted by NHðLÞ. Especially, for a vertex v in G,

NðvÞ \ VðHÞ is denoted by NHðvÞ and dHðvÞ ¼ jNHðvÞj and:

dwHðvÞ ¼X

x2NHðvÞwðvxÞ:

The minimum degree of G is denoted by �ðGÞ and �wðHÞ is defined as

minfdwGðxÞ j x 2 Hg.

Let C ¼ u1u2 � � � ujCju1 be a cycle with a fixed orientation. The segment

uiuiþ1 � � � uj is written by uiCuj. Let P ¼ u1u2 � � � ujPj be a path. The segment

uiuiþ1 � � � uj is denoted by uiPuj or P½ui; uj� and P½ui; uj� � fui; ujg by Pðui; ujÞ.A subgraph F of G is called a ðy;LÞ-fan if F has the following decomposition

F ¼Sk

i¼1 Pi, where

* k � 2,* each Pi is a path between y and vi 2 L, and* Pi \ L ¼ fvig and Pi \ Pj ¼ fyg if i 6¼ j.

Let e ¼ xy be an edge of a graph G. When we identify x with y as a new vertex

ve, we call this operation contraction of the edge e and the new graph is denoted

by G=e. Formally, G=e is a graph such that

* VðG=eÞ ¼ ðVðGÞ [ fvegÞnfx; yg* EðG=eÞ ¼ EðG� fx; ygÞ [ fvev : xv 2 EðGÞnfeg or yv 2 EðGÞnfegg

All notation and terminology not explained here are given in [5].

2. MAIN RESULTS

There are many results on the existence of long cycles. The following theorem is

a well-known result of Dirac.

Theorem A (Dirac [6]). Let G be a 2-connected graph and d an integer. If

dðvÞ � d for every vertex v in G, then G contains either a cycle of length at least

2d or a hamiltonian cycle.

And the following results show the existence of long cycles passing through

some given vertices, under the conditions of Dirac’s Theorem.

Theorem B (Groschel [8]). Let G be a 2-connected graph and d an integer.

If dðvÞ � d for every vertex v in G, then for any given vertex y in G, G contains

either a y-cycle of length at least 2d or a hamiltonian cycle.

HEAVY CYCLES IN WEIGHTED GRAPHS 95

Theorem C (Lock [10]). Let G be a 2-connected graph and d an integer. If

dðvÞ � d for every vertex v in G, then for any given vertices x and y in G, G

contains either an ðx; yÞ-cycle of length at least 2d or a hamiltonian cycle.

Theorems A and B are generalized to weighted graphs by the following two

theorems, respectively.

Theorem 1 (Bondy and Fan [3]). Let G be a 2-connected weighted graph and d

a nonnegative real number. If dwðvÞ � d for every vertex v in G, then either G

contains a cycle of weight at least 2d or every heaviest cycle in G is a

hamiltonian cycle.

Theorem 2 (Zhang et al. [13]). Let G be a 2-connected weighted graph and d a

nonnegative real number. If dwðvÞ � d for every vertex v in G, then for every

vertex y in G, either G contains a y-cycle of weight at least 2d or every heaviest

cycle in G is a hamiltonian cycle.

In this paper, we prove the following theorem, which generalizes Theorem C to

weighted graphs.

Theorem 3. Let G be a 2-connected weighted graph and d a nonnegative real

number. If dwðvÞ � d for any vertex v in G, then for every two vertices x and y in

G, either G contains an ðx; yÞ-cycle of weight at least 2d or every heaviest cycle in

G is a hamiltonian cycle.

On the other hand, the following result, which was shown by several authors

independently, gives a generalization of Theorem A in unweighted graphs. Let,

�2ðGÞ ¼ minfdðuÞ þ dðvÞ j u and v are nonadjacentg:

Theorem D (Bermond [1], Linial [9], Posa [12]). Let G be a 2-connected

graph. Then G contains either a cycle of length at least �2ðGÞ or a hamiltonian

cycle.

Enomoto [7] gave a further generalization of Theorem D as follows.

Theorem E (Enomoto [7]). Let G be a 2-connected graph and y a vertex of

G. Then G contains either a y-cycle of length at least �2ðGÞ or a hamiltonian

cycle.

And, the following result is due to Bondy et al. [2], which is a weighted

generalization of Theorem D.

Theorem 4 (Bondy et al. [2]). Let G be a 2-connected weighted graph. Then G

contains either a cycle of weight at least �w2 ðGÞ or a hamiltonian cycle.

In this paper, we also prove the following, which gives a weighted generali-

zation of Theorem E. Clearly this also generalizes Theorem 4.

Theorem 5. Let G be a 2-connected weighted graph and y a vertex of G. Then

G contains either a y-cycle of weight at least �w2 ðGÞ or a hamiltonian cycle.


3. KEY LEMMA

In this section, we prove the following lemma, which shows the existence of a

heavy fan from a given vertex to a cycle. This lemma is very useful when we

consider heavy cycles passing through some given vertices.

Lemma 1. Let G be a 2-connected weighted graph, and L;M be subgraphs of G

such that L is 2-connected and M is a component of G� L. If dwGðvÞ � d for every

vertex v in VðMÞ, then for every vertex y in VðMÞ, there exists a ðy; LÞ-fan of

weight at least d.

In our proof of Lemma 1, we use the following theorem and well-known facts.

Theorem 6 (Zhang et al. [13]). Let G be a 2-connected weighted graph and d a

nonnegative real number. Let x and z be distinct vertices of G. If dwðvÞ � d for all

v 2 VðGÞnfx; zg, then for any given vertex y of G, the graph G contains an

ðx; y; zÞ-path of weight at least d.

Fact 1. Let G be a 2-connected graph such that jGj � 4. If there exists an edge

e ¼ uv such that G=uv is not 2-connected, then the subgraph G� fu; vg is not

connected.

Fact 2. Let G be a 2-connected graph and L be a subgraph of G which is non-

separable. Then for any component H of G� L, the subgraph G½VðLÞ [ VðHÞ� is2-connected.

Proof of Lemma 1. We use induction on jMj. If M ¼ fyg, let F ¼S

v2NðyÞ vy.

Then F is a ðy;LÞ-fan such that wðFÞ ¼ dwGðyÞ � d, which is a required fan. Now

suppose jMj � 2.

Case 1. There exists t 2 NLðMÞ such that yt =2 EðGÞ or wðxtÞ > wðytÞ for somex 2 NMðtÞ.

Take a vertex x 2 NMðtÞ such that wðxtÞ is as large as possible, and make a

new graph G0 ¼ G=xt. We regard the contracted vertex as t, and let M0 be the

y-component of G0 � L. Then, it is obvious that VðM0Þ � VðMÞ.Case 1.1. G0½VðLÞ [ VðM0Þ� is 2-connected.

We first define a mapping w0G from the set of all 2-element subsets of VðGÞ to

nonnegative number set such that

* if e 2 EðGÞ, w0GðeÞ ¼ wGðeÞ, and

* if e =2 EðGÞ, w0GðeÞ ¼ 0.

Let G� ¼ G0½VðLÞ [ VðM0Þ�, and we assign weights to the edges in EðG�Þ as

follows:

a. if e 2 EðG�Þ and e is not incident to t, wG�ðeÞ ¼ wGðeÞ, and

b. if tt0 2 EðG�Þ, wG�ðtt0Þ ¼ w0Gðxt0Þ þ w0

Gðtt0Þ.


Then

* G� is 2-connected,* L� ¼ G�½VðLÞ� and M� ¼ G�½VðM0Þ� are subgraphs of G� such that L� is

2-connected and M� is a component of G� � L�,* dwG�ðvÞ � d for every vertex v in VðM�Þ (note that dwG�ðvÞ ¼ dwGðvÞ for every

vertex v 2 M�), and* jM�j < jMj.

By the induction hypothesis, we can find a ðy;L�Þ-fan F� of weight at least d in

G�. If t =2 F�, we can find a required fan F ¼ F� in G. So suppose that t 2 F�.It follows from t 6¼ y that there is only one neighbor of t in F�. We denote it by t�.

If t�x =2 EðGÞ, it holds that tt� 2 EðGÞ and wG�ðtt�Þ ¼ wGðtt�Þ. Therefore, we

can find a required fan F ¼ F� in G. If t�x 2 EðGÞ, let F ¼ F� � tt� þ t�xt.Then F is a ðy;LÞ-fan in G such that

wðFÞ ¼ wðF�Þ � wG�ðtt�Þ þ wGðt�xÞ þ wGðxtÞ¼ wðF�Þ � ðw0

Gðtt�Þ þ w0Gðxt�ÞÞ þ wGðt�xÞ þ wGðxtÞ

¼ wðF�Þ � w0Gðtt�Þ þ wGðxtÞ

� wðF�Þ � d:

Hence, F is the required fan.

Case 1.2. G0½VðLÞ [ VðM0Þ� is not 2-connected.

In this case, by Fact 2, G0 is not 2-connected, and so by Fact 1, fx; tg is a 2-cut

set of G. Clearly M� ¼ G½VðM0Þ� is the y-component of G� fx; tg, and so

NG�M�ðMÞ ¼ fx; tg.

Let H ¼ G½VðM�Þ [ fx; tg�, then Fact 2 implies that H is 2-connected. Since

VðM�Þ � VðMÞ, we have dwHðvÞ ¼ dwGðvÞ � d for every v 2 VðM�Þ, so it follows

from Theorem 6 that there exists an ðx; y; tÞ-path P1 of weight at least d in

G½M� [ fx; tg�.Since G is 2-connected and NLðM�Þ ¼ ftg, NLðM �M�Þnftg 6¼ ;. Let

t0 2 NLðM �M�Þnftg and P2 be a path joining x and t0 which does not pass

through M� (see Fig. 1). Then, F ¼ P1 [ P2 is a ðy;LÞ-fan of weight at least d.

Case 2. For every vertex t 2 NLðMÞ, yt 2 EðGÞ and wðxtÞ � wðytÞ for all

x 2 NMðtÞ.Claim 1. There exists a vertex z 2 NMðLÞnfyg and a ðy; zÞ-path P in M such

that the weight of P is at least minfdwMðzÞ; dg.

Proof. Note first that

dwMðvÞ ¼ dwGðvÞ � d for all v 2 VðMÞnNMðLÞ: ð1ÞIn the case where M is 2-connected, let z be a vertex in NMðLÞnfyg such that

dwMðzÞ � dwMðvÞ for all v 2 NMðLÞnfyg. Then with (1), we have

dwMðvÞ � minfdwMðzÞ; dg for all v 2 VðMÞnfy; zg:


Hence, Theorem 6 implies that there exists a path P of weight at least

minfdwMðzÞ; dg.

Assume that M is not 2-connected. Choose an endblock B such that

y =2 IB ¼ VðBÞnfcBg, where cB is the cutvertex of M in B. Then, there exists a

ðy; cBÞ-path P1 such that P1 \ B ¼ fcBg. Now let z be a vertex in NIBðLÞ such that

dwMðzÞ � dwMðvÞ for all v 2 NIBðLÞ: ð2ÞIf jIBj ¼ 1, then wðzcBÞ ¼ dwMðzÞ. Hence, P ¼ zcBP1y satisfies wðPÞ � dwMðzÞ. So

we may assume that jIBj � 2. Then, B is 2-connected. It follows from (1) and (2)

that

dwMðvÞ � minfdwMðzÞ; dg for all v 2 VðBÞnfcB; zg:

Then, Theorem 6 implies that there exists a ðz; cBÞ-path P2 of weight at least

minfdwMðzÞ; dg in B. Joining P1 and P2 we have the required path. &

Now we are ready to complete the proof of Case 2. Choose a vertex z and a

path P which satisfy the conditions of Claim 1. Let z0 be a neighbor of z in L and

F ¼[

v2NLðMÞnfz0gyv [ Pzz0:

Then F is a ðy;LÞ-fan such that

FIGURE 1.

wðFÞ ¼X

v2NLðMÞnfz0gwðyvÞ þ wðPÞ þ wðzz0Þ

�X

v2NLðzÞnfz0gwðzvÞ þ wðPÞ þ wðzz0Þ

� dwL ðzÞ þ minfd; dwMðzÞg� minfd; dwGðzÞg¼ d:


Hence, F is the required fan. This completes the proof of Case 2 and the proof of

Lemma 1. &

4. PROOFS OF THEOREMS 3 AND 5

First we prove Theorem 3. Before proving this, we give alternative proofs of

Theorems 1 and 2, using Lemma 1.

Proof of Theorem 1. Let G be a weighted graph satisfying the conditions of

Theorem 1. Assume that there exists a heaviest cycle C in G which is not a

hamiltonian cycle and wðCÞ < 2d. Now take a vertex y 2 G� C. Then, from

Lemma 1, we have a ðy;CÞ-fan F of weight � d. Let F \ C ¼ fv1; v2; . . . ; vkgwhere vi are in order around C. And let Ci ¼ viPiyPiþ1viþ1Cvi, where Pi is the

path connecting vi and y in F. Then,Pk

i¼1 wðCiÞ ¼ ðk � 1ÞwðCÞ þ 2wðFÞ �ðk � 1ÞwðCÞ þ 2d > kwðCÞ. Hence, among these k cycles, there must be one whose

weight is greater than wðCÞ, contradicting that C is a heaviest cycle in G. &


Theorem 2. Assume that there exists a heaviest cycle C in G which is not a

hamiltonian cycle. Then from Theorem 1, wðCÞ � 2d. If y 2 C, there is nothing

to prove, so assume that y =2 C. It follows from Lemma 1 that there is a ðy;CÞ-fan

F of weight � d. Now take Ci as in the proof of Theorem 1. Then, each Ci is a

y-cycle, andPk

i¼1 wðCiÞ ¼ ðk � 1ÞwðCÞ þ 2wðFÞ � 2kd. Hence, among these k

y-cycles, there must be one of weight at least 2d. &


Theorem 3. If there is a heaviest cycle which is not a hamiltonian cycle, then

Theorem 2 implies that there exists some cycle of weight �2d which contains

either x or y. Let C be the heaviest one among these cycles. Without loss of

generality, we can assume that C contains x. If y 2 C, there is nothing to prove, so

assume that y =2 C. It follows from Lemma 1 that there is a ðy;CÞ-fan F of weight

�d. Now take Ci as in the proof of Theorem 1. Then, each Ci is a y-cycle, andPki¼1 wðCiÞ ¼ ðk � 1ÞwðCÞ þ 2wðFÞ � 2kd. If all of these k cycles contain x,

there must be one of weight at least 2d, which is the required cycle. Otherwise,

there is only one cycle Cl which does not contain x. But from the choice of C,

wðClÞ � wðCÞ. Hence,P

i6¼l wðCiÞ � ðk � 2ÞwðCÞ þ 2wðFÞ � 2ðk � 1Þd, so one

of them is an ðx; yÞ-cycle of weight at least 2d. &

Next, we prove Theorem 5. Now we prepare a lemma which is used in the

proof of Theorem 5. Zhu [14] showed that a 2-connected graph G contains a cycle

of length at least 2ð�2ðGÞ � �ðGÞÞ or a hamiltonian cycle. However, we cannot

give its weighted generalization. Let G be the complete bipartite graph Kk;kþ1

with partite set V1 of order k. Let u 2 V1, and we assign weight zero to every edge

incident with u, and suppose other edges have weight one. Then �w2 ðGÞ ¼ k þ 1


and �wðGÞ ¼ 0, and the weight of a heaviest cycle is 2k � 2 and G is not

hamiltonian. However, modifying the proof of Theorem 4, easily we can obtain

the following fact.

Lemma 2. If G is a 2-connected weighted graph, then there is a cycle C of

weight at least maxf�w2 ðGÞ; 2ð�w2 ðGÞ � �wðG� CÞÞg or a hamiltonian cycle.

Proof. Let P ¼ u1u2 � � � up be a heaviest path in all longest paths in G. Let

el ¼ ul�1ul and e0l ¼ u1ul for all ul 2 Nðu1Þ, and fl ¼ ululþ1 and f 0l ¼ ulup for all

ul 2 NðupÞ. Suppose G is not hamiltonian. Then ful j ulþ1 2 Nðu1Þg \ NðupÞ ¼ ;and so fel j ul 2 Nðu1Þg \ f fk j uk 2 NðupÞg ¼ ; as P is longest. Because the

weight of P is at least the weights of the paths P� el þ e0l and P� fl þ f 0l , we

have wðelÞ � wðe0lÞ and wð flÞ � wð f 0l Þ.Let s ¼ maxfl j ul 2 Nðu1Þg and t ¼ minfl j ul 2 NðupÞg. If s > t, then there

exist ui 2 Nðu1Þ and uj 2 NðupÞ such that neither u1 nor up has neighbors in

Pðuj; uiÞ. See Figure 2. Then the cycle C ¼ u1PujupPuiu1 contains every edge in:

fel j ul 2 Nðu1Þnuig [ ffl j ul 2 NðupÞnujg [ fe0i; f 0j g ð3Þ

and so Nðu1Þ [ NðupÞ � VðCÞ. Therefore, both of dwðu1Þ and dwðupÞ are at

least �w2 ðGÞ � �wðG� CÞ and the following inequalities hold because

felgl \ ffkgk ¼ ;.

wðCÞ �X

ul2Nðu1ÞnuiwðelÞ þ wðe0iÞ þ

X

ul2NðupÞnujwð flÞ þ wð f 0j Þ

� dwðu1Þ þ dwðupÞ � maxf�w2 ðGÞ; 2ð�w2 ðGÞ � �wðG� CÞÞg: ð4Þ

If s ¼ t, then there is a path Q joining ui0 2 Pðu1; usÞ and uj0 2 Pðus; upÞ which

is internally disjoint to P as G is 2-connected. Let i ¼ minfl > i0 j ul 2 Nðu1Þgand j ¼ maxfl < j0 j ul 2 NðupÞg. See Figure 3. Then the cycle C ¼ u1Pui0Quj0

PupujPuiu1 contains every edge in (3), and so the inequalities (4) hold.

Suppose s < t. By Perfect’s theorem [11], there are two vertex disjoint paths

Q1 and Q2 joining P½u1; us� and P½ut; up� such that us and ut are ends of Q1 or Q2,

and both of Q1 and Q2 are internally disjoint to P½u1; us� [ P½ut; up�. Let

fui0 ; us; ut; uj0g be the set of all the ends of Q1 and Q2 such that i0 < s and j0 > t.

Let i ¼ minfl > i0 j ul 2 Nðu1Þg and j ¼ maxfl < j0 j ul 2 NðupÞg. Then the cycle:

C ¼ P½u1; ui0 � [ P½ui; us� [ P½ut; uj� [ P½uj0 ; up� [ Q1 [ Q2 [ fe0i; f 0j g

contains every edge in (3), and thus the inequalities (4) hold. &

FIGURE 2.


Now we are ready to prove Theorem 5.

Proof of Theorem 5. Assume that G is not hamiltonian. Then by Lemma 2,

there is a cycle C of weight at least maxf�w2 ðGÞ; 2ð�w2 ðGÞ � �wðG� CÞÞg.

If y 2 C, there is nothing to prove, so assume that y =2 C. Let d ¼ �wðG� CÞ.It follows from Lemma 1 that there is a ðy;CÞ-fan F of weight � d. Now take Ci

as in the proof of Theorem 1. Then, each Ci is a y-cycle, and

Xk

i¼1

wðCiÞ ¼ ðk � 1ÞwðCÞ þ 2wðFÞ

� ðk � 1ÞwðCÞ þ 2d

¼ ðk � 2ÞwðCÞ þ wðCÞ þ 2d

� ðk � 2Þ�w2 ðGÞ þ 2ð�w2 ðGÞ � dÞ þ 2d

¼ k�w2 ðGÞ:

Hence, one of them is a y-cycle of weight at least �w2 ðGÞ. &

ACKNOWLEGMENT

This work was done when Shenggui Zhang was visiting Nihon University,

supported by NSFC (No. 10101021).

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