a degree sum condition for longest cycles in 3-connected graphs

A Degree Sum Conditionfor Longest Cycles in3-Connected Graphs

Tomoki Yamashita

DEPARTMENT OF MATHEMATICSSCHOOL OF DENTISTRY, ASAHI UNIVERSITY

1851 HOZUMI, GIFU 501–0296JAPAN

E-mail: [email protected]

Received July 6, 2005; Revised July 19, 2006

Published online 27 October 2006 in Wiley InterScience(www.interscience.wiley.com).DOI 10.1002/jgt.20210

Abstract: For a graph G, we denote by dG(x) and κ(G) the degree of avertex x in G and the connectivity of G, respectively. In this article, we showthat if G is a 3-connected graph of order n such that dG(x) + dG(y) + dG(z) ≥d for every independent set {x, y, z}, then G contains a cycle of length atleast min{d − κ(G), n}. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 277–283, 2007

Keywords: degree sum; connectivity; longest cycle; circumference

1. INTRODUCTION

In this article, we consider only finite undirected graphs without loops or multipleedges. Let G be a graph and X ⊂ V (G). We denote the degree of a vertex x in G

by dG(x). We define δ(X) := min{dG(x) : x ∈ X}. We denote by G[X] an inducedsubgraph by X, and by α(X) the maximum number of pairwise nonadjacent verticesin G[X]. If G[X] is not complete, we denote by κ(X) the minimum cardinality ofa set of vertices of G separating two vertices of X. If G[X] is complete, we defineκ(X) = |X| − 1. Let

Journal of Graph Theory© 2006 Wiley Periodicals, Inc.

277

278 JOURNAL OF GRAPH THEORY

σk(X) =min

{∑x∈S

dG(x) : S is an independent set of G[X] with |S| = k

}if α(X) ≥ k

+∞ if α(X) < k.

We simply write δ, α, κ, and σk instead of δ(V (G)), α(V (G)), κ(V (G)), andσk(V (G)), respectively.

The following two results are well-known in hamiltonian graph theory.

Theorem 1 (Ore [7]). Let G be a graph of order n ≥ 3. If σ2 ≥ n, then G ishamiltonian.

Theorem 2 (Chvatal and Erdos [4]). Let G be a graph of order n ≥ 3. If α ≤ κ,then G is hamiltonian.

For X ⊂ V (G) with |X| ≥ k, let

�k(X) = max

{∑x∈S

dG(x) : S is a subset of X with |S| = k

}.

We define

σkr (X) ={min{�r(S) : S is an independent set of G[X] with |S| = k} if α(X) ≥ k

+∞ if α(X) < k.

We also write σkr instead of σk

r (V (G)). Recently, we gave a common generaliza-tion of Theorems 1 and 2 involving this parameter.

Theorem 3 ([9]). Let G be a connected graph on n vertices. If σκ+12 ≥ n, then G

is hamiltonian.

The following theorems are generalizations of Theorem 3 in different directions.

Theorem 4 ([9]). Let G be a 2-connected graph on n vertices, and X a subset ofV (G). If σ

κ(X)+12 (X) ≥ n, then G contains a cycle passing through X.

Theorem 5 ([9]). Let G be a 2-connected graph on n vertices. Then G containsa cycle of length at least min{σκ+1

2 , n}.On the other hand, Bauer et al. (1989) gave a σ3 condition that guarantees the

existence of a hamiltonian cycle.

Theorem 6 (Bauer et al. [1]). Let G be a 2-connected graph on n vertices. Ifσ3 ≥ n + κ, then G is hamiltonian.

Journal of Graph Theory DOI 10.1002/jgt

DEGREE SUM CONDITION IN 3-CONNECTED GRAPHS 279

Broersma et al. (1997) showed a generalization of Theorem 6 in terms of cyclespassing through specified vertices.

Theorem 7 (Broersma et al. [3]). Let G be a 2-connected graph on n vertices,and X a subset of V (G). If σ3(X) ≥ n + min{κ(X), δ(X)}, then G contains a cyclepassing through X.

In this article, we prove the following result concerning the length of a longestcycle.

Theorem 8. Let G be a 3-connected graph on n vertices. Then G contains a cycleof length at least min{σ3 − κ, n}.

Several results are known for the length of a longest cycle in a 3-coonectedgraph. Let σ3 = min{∑3

i=1 dG(xi) − | ⋂3i=1 N(xi)| : {x1, x2, x3} is an independent

set of G}, and let NC = min{|N(u) ∪ N(v)| : uv �∈ E(G), u �= v}.Theorem 9 (Wei [8]). Let G be a 3-connected graph on n vertices. Then G

contains a cycle of length at least min{σ3, n}.Theorem 10 (Liu [6]). Let G be a 3-connected graph on n vertices. Then G

contains a cycle of length at least min{3(NC + 1)/2, n}.Theorem 8 is best possible and is not weaker than Theorems 5, 9, and 10. Let

k, m, n be integers with 3 ≤ k < m and 2 ≤ n < m − k + 1. We consider the graphG obtained from (Km + mK1) ∪ Kn by joining k vertices of Km and each vertex ofKn. Then κ = k, σ3 = 2m + n + κ − 1, σκ+1

2 = 2m, σ3 = 2m, NC = m + n − 1,and c(G) = 2m + n − 1 = σ3 − κ > max{σκ+1

2 , σ3, 3(NC + 1)/2}, where c(G) isthe length of a longest cycle in G.

In Theorem 8, the condition “3-connected” cannot be weakened to “2-connected.” Let G = (Kp ∪ Kq ∪ Kr) + K2 (2 ≤ p ≤ q ≤ r). Then κ = 2, σ3 −κ = p + q + r + 1 and c(G) = q + r + 2 < min{σ3 − κ, |V (G)|}.

Fraisse and Jung (1989) showed the following result. A cycle C of a graph G issaid to be a dominating cycle if V (G \ C) is an independent set.

Theorem 11 (Fraisse and Jung [5, Corollary 20]). Let G be a 3-connected graphon n vertices. Then any longest cycle is dominating or G contains a cycle of lengthat least min{σ3 − 3, n}.

By combining Theorem 11 and the following lemma, we shall prove Theorem 8.

Lemma 1. Let G be a 2-connected graph on n vertices. If any longest cycle isdominating, then G contains a cycle of length at least min{σ3 − κ, n}.

2. PROOF OF LEMMA 1

For standard graph-theoretic terminology not explained in this article, we refer thereader to [2]. Let G be a graph and H be a subgraph of G, and let x ∈ V (G) andX ⊂ V (G). We denote by NG(x) and NG(X) the neighborhood in G of x and the



set of vertices in V (G \ X) which are adjacent to some vertex in X, respectively.We define NH (x) := NG(x) ∩ V (H) and dH (x) := |NH (x)|. Furthermore, we defineNH (X) := NG(X) ∩ V (H). If there is no fear of confusion, we often identify H withits vertex set V (H). For example, we often write G \ H instead of G \ V (H).

We write a cycle C with a given orientation by−→C . For x, y ∈ V (C), we denote

by C[x, y] a path from x to y on−→C . The reverse sequence of C[x, y] is denoted

by←−C [y, x]. We define C(x, y) = C[x, y] \ {x, y}. For x ∈ V (C), we denote the

successor and the predecessor of u on−→C by x+ and x−, respectively. For a cycle−→

C and X ⊂ V (C), we define X+ := {x+ : x ∈ X} and X− := {x− : x ∈ X}.A path P connecting x and y is denoted by P[x, y]. We say a path P[x, y] is

maximal if NG(x) ∪ NG(y) ⊂ V (P). For a subgraph H of G, a path P[x, y] is calledan H-path if V (P) ∩ V (H) = {x, y} and E(H) ∩ E(P) = ∅.

To prove Lemma 1, we use the following two lemmas.

Lemma 2 (Bondy). Let G be a 2-connected graph. If P[u, v] is a maximal pathof G, then there exists a cycle of length at least min{dG(u) + dG(v), n}.

The following lemma is easy to prove, and so we omit the proofs.

Lemma 3. Let G be a graph, let C be a longest cycle in G, let H be a componentof G \ C and let u, v ∈ NC(H) with u �= v. Then the following statements hold.

(1) u+ �∈ NC(H).(2) There exists no C-path connecting u+ and v+.(3) If there exists a C-path connecting u+ and w ∈ C(u+, v+), then there exists

no C-path connecting v+ and w−.

Proof of Lemma 1. Let S be a κ-cut set of G, let H1, . . . , Hm be a component ofG \ S and let Vi := V (Hi) for 1 ≤ i ≤ m. Since S is a cut set, m ≥ 2 and so, withoutloss of generality, we may assume that Vi �= ∅ for i = 1, 2. Let C be a longest cycle.Then C is a dominating cycle, by the hypothesis. Hence note that NG(v) = NC(v)for every v ∈ V (G − C). If V (G \ C) = ∅, then we obtain the conculsion, and sosuppose that V (G \ C) �= ∅. Let x0 ∈ V (G \ C) and let T := NG(x0) = NC(x0) ={v1, . . . , vk}. Let xi := v+

i for 1 ≤ i ≤ k.

Case 1. dG(x0) = κ.

If NG\C(xi) �= ∅ for i = 1, 2, then let x′i ∈ NG\C(xi); otherwise let x′

i = xi.Then P = x′

1−→C [x1, v2]x0

←−C [v1, x2]x′

2 is a maximal path, since NG(x′i) = NC(x′

i) ⊂V (C) ⊂ V (P). By Lemma 2, there exists a cycle C′ of length at least min{dG(x′

1) +dG(x′

2), n}. By Lemma 3(1) and (2), {x0, x′1, x

′2} is an independent set of order

three. Hence we obtain |C′| ≥ min{dG(x′1) + dG(x′

2), n} ≥ min{σ3 − dG(x0), n} =min{σ3 − κ, n}. This completes the proof of Case 1.

Case 2. dG(x0) ≥ κ + 1.



If NG\C(v) \ S �= ∅, then let v∗ ∈ NG\C(v) \ S; otherwise let v∗ = v. Notethat v∗ ∈ Vi if v ∈ Vi for some i. Let A := {xi : vi, xi �∈ S} and B := {x ∈ A :NG\C(x) ∩ S �= ∅}. Then, A ⊂ T + and B ⊂ A.

Claim 1. Let X ⊂ S \ (T ∪ T +). Then |A| ≥ |X| + 1.

Proof. Since dG(x0) ≥ κ + 1, |T +| − |S| = |T | − |S| ≥ 1. Hence |A| ≥|T +| − (|S| − |X|) = |T +| − |S| + |X| ≥ 1 + |X|. �

By Lemma 3(2), NG\C(xi) ∩ NG\C(xj) = ∅ for 1 ≤ i �= j ≤ k. Hence we have|NG\C(B) ∩ S| ≥ |B|. By applying Claim 1 as X = NG\C(B) ∩ S, we obtain |A| ≥|NG\C(B) ∩ S| + 1 ≥ |B| + 1 and so |A| − |B| ≥ 1. Since B ⊂ A, we have |A \B| ≥ 1, that is, A \ B �= ∅. Let x1 ∈ A \ B. Without loss of generality, we mayassume that x1 ∈ V1. Then, by the definition of A, v1 ∈ V1 and x∗

1 ∈ V1 by thedefinition of x∗

1.

Case 2.1.⋃m

i=2 Vi ⊂ T .

Then x0 ∈ S. By Lemma 3(1), x0 �∈ NG\C(B). By applying Claim 1 as X =(NG\C(B) ∩ S) ∪ {x0}, we have |A \ B| ≥ 2. Hence there exists, without loss ofgenerality, say x2 ∈ A \ B with x1 �= x2, and by Lemma 3(2), x∗

1 �= x∗2 and x∗

1x∗2 �∈

E(G). Let C1 := C[x1, v2] and C2 := C[x2, v1]. By Lemma 3(2), NC1 (x∗1)− ∩ (T \

{v1}) = ∅. Hence NC1 (x∗1)− ∩ V2 = ∅, since V2 ⊂ T and v1 /∈ V2. If x∗

1 ∈ V (G \ C)then x+

1 �∈ NC(x∗1), that is, x1 �∈ NC(x∗

1)−. By the assumption of Case 2.1, x2 ∈ V1

and so x∗2 ∈ V1. This yields NG(x∗

2) ∩ V2 = ∅. By Lemma 3(2), x1 �∈ NG(x∗2). Thus,

we obtain

NC1 (x∗1)− ∪ NC1 (x

∗2) ⊂

{C1 \ V2 if x∗

1 ∈ V (C)

(C1 \ (V2 ∪ {x1})) ∪ {v1} if x∗1 ∈ V (G \ C).

By Lemma 3(2) and (3), NC1 (x∗1)− ∩ NC1 (x

∗2) = ∅. Hence, since |C1 \ V2| =

|((C1 \ V2) \ {x1}) ∪ {v1}|, we have dC1 (x∗1) + dC1 (x

∗2) ≤ |C1 \ V2|. By symmetry,

dC2 (x∗1) + dC2 (x

∗2) ≤ |C2 \ V2|. Since V2 ⊂ V (C), we deduce that

dG(x∗1) + dG(x∗

2) = dC(x∗1) + dC(x∗

2)

≤ |C1 \ V2| + |C2 \ V2|≤ |C1| − |C1 ∩ V2| + |C2| − |C2 ∩ V2|≤ |C| − |V2|.

Let y1 ∈ V2. Then dG(y1) ≤ |V2| + |S| − 1. Since x∗1, x

∗2 ∈ V1 and y1 ∈ V2,

{x∗1, x

∗2, y1} is an independent set of order three. Thus σ3 ≤ dG(x∗

1) + dG(x∗2) +

dG(y1) ≤ |C| + |S| − 1 = |C| + κ − 1 and so |C| ≥ σ3 − κ + 1.

Case 2.2.⋃m

i=2 Vi �⊂ T .



Let y2 ∈ ⋃mi=2 Vi \ T . Choose y2 ∈ ⋃m

i=2 Vi ∩ T + if⋃m

i=2 Vi ∩ T + �= ∅. Withoutloss of generality, we may assume that y2 ∈ V2. Since y2 �∈ T , we get y∗

2 �= x0 andx0y

∗2 �∈ E(G). Hence {x0, x

∗1, y

∗2} is an independent set of order three.

Since x1 ∈ A \ B, NG\C(x1) ∩ S = ∅. By the definition of v∗ and the fact thatC is dominating, NG\C(x∗

1) = ∅ and NG\C(y∗2) ⊂ S. By Lemma 3(2), NC(x∗

1) ∩(T + \ {x1}) = ∅. Note that x1 �∈ NG(y∗

2) because x1 ∈ V1 and y∗2 ∈ V2. Hence, since

x∗1 ∈ V1 and y∗

2 ∈ V2, we obtain NG(x∗1) ∩ NG(y∗

2) ⊂ (C \ T +) ∩ S. We define

Y :={

T + \ S if x∗1 ∈ V (C)

(T + \ (S ∪ {x1})) ∪ {v1} if x∗1 ∈ V (G \ C).

It follows from Lemma 3(1) and (2) that NC(x∗1) ∩ Y = ∅. Note that v1 �∈ NC(y∗

2)since v1 ∈ V1 and y∗

2 ∈ V2. First, suppose that y2 �∈ T +. Then, by the choice ofy2, T + \ S ⊂ V1. Hence NC(y∗

2) ∩ Y = ∅. Thus, we have NG(x∗1) ∪ NG(y∗

2) ⊂ (C \Y ) ∪ ((G \ C) ∩ S). Next, suppose that y2 ∈ T +. We define

Z :={

Y if y∗2 ∈ V (C)

Y \ {y2} ∪ {y+2 } if y∗

2 ∈ V (G \ C).

By Lemma 3(1) and (2), NC(y∗2) ∩ Z = ∅. If y∗

2 ∈ V (G \ C) then y+2 �∈ NG(x∗

1),otherwise C[x1, v2]x0

←−C [v1, y

+2 ]x1 is a longest cycle but not dominating. This

yields NC(x∗1) ∩ Z = ∅. Therefore NG(x∗

1) ∪ NG(y∗2) ⊂ (C \ Z) ∪ ((G \ C) ∩ S).

Since |T + \ S| = |Y | = |Z|, we obtain

dG(x∗1) + dG(y∗

2) ≤ |C| − |T + \ S| + |(G \ C) ∩ S| + |(C \ T +) ∩ S|≤ |C| − |T + \ S| + |(G \ C) ∩ S| + |C ∩ S| − |T + ∩ S|= |C| + |S| − |T +|= |C| + κ − dG(x0),

and so |C| ≥ dG(x0) + dG(x∗1) + dG(y∗

2) − κ ≥ σ3 − κ. �

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[1] D. Bauer, H. J. Broersma, R. Li, and H. J. Veldman, A generalization of a resultof Haggkvist and Nicoghossian, J Combin Theory Ser B 47 (1989), 237–243.

[2] J. A. Bondy, “Basic graph theory—Paths and cycles,” Handbook of Combina-torics, Vol. I, Elsevier, Amsterdam, 1995, pp. 5–110.

[3] H. J. Broersma, H. Li, J. Li, F. Tian, and H. J. Veldman, Cycles through subsetswith large degree sums, Discrete Math 171 (1997), 43–54.

[4] V. Chvatal and P. Erdos, A note on hamiltonian circuits, Discrete Math 2 (1972),111–113.



[5] P. Fraisse and H. A. Jung, “Longest cycles and independent sets in k-connectedgraphs,” Recent Studies in Graph Theory, V. R. Kulli, (Editor), Vischwa Internat.Publ. Gulbarga, India, 1989, pp. 114–139.

[6] X. Liu, Lower bounds of length of longest cycles in graphs involving neighbor-hood unions, Discrete Math 169 (1997), 133–144.

[7] O. Ore, Note on Hamilton circuits, Amer Math Monthly 67 (1960), 55.[8] B. Wei, Longest cycles in 3-connected graphs, Discrete Math 170 (1997), 195–

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a degree sum condition for longest cycles in 3-connected graphs

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