Contractible subgraphs, Thomassen’s conjecture and the dominating cycle conjecture for snarks

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is Hamiltonian), by Thomassen (every 4-connected line graph is Hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edge-coloring or a dominating cycle), which are known to be equivalent, are equivalent to the statement that every snark (i.e. a cyclically 4-edge-connected cubic graph of girth at least five that is not 3-edge-colorable) has a dominating cycle.We use a refinement of the contractibility technique which was introduced by Ryjá?ek and Schelp in 2003 as a common generalization and strengthening of the reduction techniques by Catlin and Veldman and of the closure concept introduced by Ryjá?ek in 1997.     

Hamiltonicity in claw-free graphs through induced bulls

In the present paper we show that if G is a 2-connected claw-free graph such that the vertices of degree 1 of every induced bull have a common neighbour in G then G is hamiltonian. This statement was originally conjectured by H.J. Broersma and H.J. Veldman     

Degree conditions restricted to induced paths for hamiltonicity of claw-heavy graphs

Broersma and Veldman proved that every 2-connected claw-free and P ₆-free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected claw-heavy and P ₆-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and P ₆-o-heavy but not hamiltonian. In this paper, we further give some Ore-type degree conditions restricting to induced copies of P ₆ of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.     

Contractible Subgraphs,Thomassen's Conjecture and the Dominating Cycle Conjecture for Snarks

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is hamiltonian), by Thomassen (every 4-connected line graph is hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edge-coloring or a dominating cycle), which are known to be equivalent, are equivalent with the statement that every snark (i.e. a cyclically 4-edge-connected cubic graph of girth at least five that is not 3-edge-colorable) has a dominating cycle.We use a refinement of the contractibility technique which was introduced by Ryjáček and Schelp in 2003 as a common generalization and strengthening of the reduction techniques by Catlin and Veldman and of the closure concept introduced by Ryjáček in 1997.     

On Hamiltonicity of 3-Connected Claw-Free Graphs

Lai, Shao and Zhan (J Graph Theory 48:142–146, 2005) showed that every 3-connected N ₂-locally connected claw-free graph is Hamiltonian. In this paper, we generalize this result and show that every 3-connected claw-free graph G such that every locally disconnected vertex lies on some induced cycle of length at least 4 with at most 4 edges contained in some triangle of G is Hamiltonian. It is best possible in some sense.     

Hamiltonian connected hourglass free line graphs

Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K₅-E(C₄), where C₄ is a cycle of length 4 in K₅. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected.     

直径不超过2的2连通无爪图是哈密顿图

不包含2K_2的图是指不包含一对独立边作为导出子图的图.Kriesell证明了所有4连通的无爪图的线图是哈密顿连通的.本文证明了如果图G不包含2K_2并且不同构与K_2,P_3和双星图,那么线图L(G)是哈密顿图,进一步应用由Ryjá(?)ek引入的闭包的概念,给出了直径不超过2的2连通无爪图是哈密顿图这个定理的新的证明方法.     

On hamiltonicity of 2-connected claw-free graphs

A graph G has the hourglass property if every induced hourglass S (a tree with a degree sequence 22224) contains two non-adjacent vertices which have a common neighbor in G - V(S).For an integer k ≥ 4,...     

Longest Cycles in Almost Claw-Free Graphs

Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we prove several properties on longest cycles in almost claw-free graphs. In particular, we show the following two results.? (1) Every 2-connected almost claw-free graph on n vertices contains a cycle of length at least min {n, 2δ+4} and the bound 2δ+ 4 is best possible, thereby fully generalizing a result of Matthews and Sumner.? (2) Every 3-connected almost claw-free graph on n vertices contains a cycle of length at least min {n, 4δ}, thereby fully generalizing a result of MingChu Li. Received: September 17, 1996 Revised: September 22, 1998     

Hamilton cycles in claw-heavy graphs

A graph G on n≥3 vertices is called claw-heavy if every induced claw (K_1,3) of G has a pair of nonadjacent vertices such that their degree sum is at least n. In this paper we show that a claw-heavy graph G has a Hamilton cycle if we impose certain additional conditions on G involving numbers of common neighbors of some specific pair of nonadjacent vertices, or forbidden induced subgraphs. Our results extend two previous theorems of Broersma, Ryjá?ek and Schiermeyer [H.J. Broersma, Z. Ryjá?ek, I. Schiermeyer, Dirac’s minimum degree condition restricted to claws, Discrete Math. 167-168 (1997) 155-166], on the existence of Hamilton cycles in 2-heavy graphs.     

2-连通无爪图的连通因子

若图G不含有同构于K1,3的导出子图,则称G为一个无爪图.令a和b是两个整数满足2≤a≤b.本文证明了若G是一个含有[a,b]因子的2连通无爪图,则G有一个连通的[a,b 1]因子.     

INDEPENDENT-SET-DELETABLE FACTOR-CRITICAL POWER GRAPHS

It is said that a graph G is independent-set-deletable factor-critical (in short, ID-factor-critical), if, for every independent set 7 which has the same parity as |V(G)|, G-I has a perfect matching. A graph G is strongly IM-extendable, if for every spanning supergraph H of G, every induced matching of H is included in a perfect matching of H. The k-th power of G, denoted by Gk, is the graph with vertex set V(G) in which two vertices are adjacent if and only if they have distance at most k in G. ID-factor-criticality and IM-extendability of power graphs are discussed in this article. The author shows that, if G is a connected graph, then G3 and T(G) (the total graph of G) are ID-factor-critical, and G4 (when |V(G)| is even) is strongly IM-extendable; if G is 2-connected, then D2 is ID-factor-critical.     

Degree conditions on claws and modified claws for hamiltonicity of graphs

Ore presented a degree condition involving every pair of nonadjacent vertices for a graph to be hamiltonian. Fan [New sufficient conditions for cycles in graphs, J. Combin. Theory Ser. B 37 (1984) 221-227] showed that not all the pairs of nonadjacent vertices are required, but only the pairs of vertices at the distance two suffice. Bedrossian et al. [A generalization of Fan's condition for hamiltonicity, pancyclicity, and hamiltonian connectedness, Discrete Math. 115 (1993) 39-50] improved Fan's result involving the pairs of vertices contained in an induced claw or an induced modified claw. On the other hand, Matthews and Sumner [Longest paths and cycles in K_1,3-free graphs, J. Graph Theory 9 (1985) 269-277] gave a minimum degree condition for a claw-free graph to be hamiltonian. In this paper, we give a new degree condition in an induced claw or an induced modified claw ensuring the hamiltonicity of graphs which extends both results of Bederossian et al. and Matthews and Sumner.     

Connected even factors in claw-free graphs

A connected even [2,2s]-factor of a graph G is a connected factor with all vertices of degree i (i=2,4,…,2s), where s?1 is an integer. In this paper, we show that every supereulerian K_1,s-free graph (s?2) contains a connected even [2,2s-2]-factor, hereby generalizing the result that every 4-connected claw-free graph has a connected [2,4]-factor by Broersma, Kriesell and Ryjacek.     

无爪图中的最长圈

本文证明了:n阶3—连通无爪图G中的最长圈的长至少为min{4k—5,n},这里k是G的最小度.     

无爪图中不相邻子图P_4和K_1的度和条件下的哈密尔顿性

研究子图的度和图的哈密尔顿性的关系,证明图~$G$ 是一个~$n$ 阶~3-\,连通无爪图且最小度~$\delta(G)\geq4$, 如果图~$G$ 中任意两个分别同构于~$P_4$, $K_1$ 的不相邻子图~$H_1$, $H_2$ 满足~$d(H_1)+d(H_2)\geq n$, 则图~$G$ 是哈密尔顿连通.     

3-连通3-正则图中的可序子集

图G的k元点集X={x1,x2,…,xk}被称为G的k-可序子集,如果X的任意排列都按序排在G的某个圈上.称G是k-可序图,如果G的每一个k元子集都是G的k-可序子集.称G为k-可序Hamilton图,如果X的任意排列都位于G的Hamilton圈上.研究了3-连通3-正则图的可序子集的存在性问题.     

P3-支配图哈密尔顿性的两个充分条件

在文献[3]中介绍了一个新的图类-P3-支配图.这个图类包含所有的拟无爪图,因此也包含所有的无爪图.在本文中,我们证明了每一个点数至少是3的三角形连通的P3-支配图是哈密尔顿的,但有一个例外图K1,1,3.同时,我们也证明了k-连通的(k≥2)的P3-支配图是哈密尔顿的,如果an(G)≤k,但有两个例外图K1,1,3 and K2,3.     

Circumferences of regular claw-free graphs

A known result obtained independently by Fan and Jung is that every 3-connected k-regular graph on n vertices contains a cycle of length at least min{3k,n}. This raises the question of how much can be said about the circumferences of 3-connected k-regular claw-free graphs. In this paper, we show that every 3-connected k-regular claw-free graph on n vertices contains a cycle of length at least min{6k-17,n}.     

关于图是(r,n)-临界图的一个邻域条件

设n，m和r是满足r≥2，n≥0，m≥3的整数，且当r是奇数时，假设r≥m-1．称一个图为K1,m-free，如果它不包含以Kt,m为导出的子图．称一个图G为一个（r,n）-临界图，如果在删去G的任意n个点后，剩下G的子图都有一个r-因子,设G是一个Kl,m-free的（n＋1）-连通图，且阶为｜G｜以及r（｜G｜≥n）是偶数,证明了：如果G的最小度至少是r＋n＋m-1，阶｜G｜≥8r5＋n，并且对V（G）的任意独立点集{x1，x2}都有｜NG（x1）∪NG（x2）｜≥（｜G｜＋n）／2，那么G是一个（r,n）-临界图．关于G的最小度和｜NG（x1）∪NG（X2）｜的下界是紧的。