Line graphs of multigraphs and Hamilton‐connectedness of claw‐free graphs

We introduce a closure concept that turns a claw‐free graph into the line graph of a multigraph while preserving its (non‐)Hamilton‐connectedness. As an application, we show that every 7‐connected claw‐free graph is Hamilton‐connected, and we show that the well‐known conjecture by Matthews and Sumner (every 4‐connected claw‐free graph is hamiltonian) is equivalent with the statement that every 4‐connected claw‐free graph is Hamilton‐connected. Finally, we show a natural way to avoid the non‐uniqueness of a preimage of a line graph of a multigraph, and we prove that the closure operation is, in a sense, best possible. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:152‐173, 2011     

On stability of Hamilton‐connectedness under the 2‐closure in claw‐free graphs

It is easy to characterize chordal graphs by every k‐cycle having at least f(k) = k ? 3 chords. I prove new, analogous characterizations of the house‐hole‐domino‐free graphs using f(k) = 2?(k ? 3)/2?, and of the graphs whose blocks are trivially perfect using f(k) = 2k ? 7. These three functions f(k) are optimum in that each class contains graphs in which every k‐cycle has exactly f(k) chords. The functions 3?(k ? 3)/3? and 3k ? 11 also characterize related graph classes, but without being optimum. I consider several other graph classes and their optimum functions, and what happens when k‐cycles are replaced with k‐paths. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:137‐147, 2011     

Hamilton connectivity of line graphs and claw‐free graphs

Let G be a graph and let V₀ = {ν∈ V(G): d_G(ν) = 6}. We show in this paper that: (i) if G is a 6‐connected line graph and if |V₀| ≤ 29 or G[V₀] contains at most 5 vertex disjoint K₄'s, then G is Hamilton‐connected; (ii) every 8‐connected claw‐free graph is Hamilton‐connected. Several related results known before are generalized. © 2005 Wiley Periodicals, Inc. J Graph Theory     

A Closure for 1‐Hamilton‐Connectedness in Claw‐Free Graphs

A graph G is 1‐Hamilton‐connected if is Hamilton‐connected for every vertex . In the article, we introduce a closure concept for 1‐Hamilton‐connectedness in claw‐free graphs. If is a (new) closure of a claw‐free graph G, then is 1‐Hamilton‐connected if and only if G is 1‐Hamilton‐connected, is the line graph of a multigraph, and for some , is the line graph of a multigraph with at most two triangles or at most one double edge. As applications, we prove that Thomassen's Conjecture (every 4‐connected line graph is hamiltonian) is equivalent to the statement that every 4‐connected claw‐free graph is 1‐Hamilton‐connected, and we present results showing that every 5‐connected claw‐free graph with minimum degree at least 6 is 1‐Hamilton‐connected and that every 4‐connected claw‐free and hourglass‐free graph is 1‐Hamilton‐connected.     

Hadwiger's conjecture for quasi‐line graphs

A graph G is a quasi‐line graph if for every vertex v ∈ V(G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. Hadwiger's conjecture states that if a graph G is not t‐colorable then it contains K_{t + 1} as a minor. This conjecture has been proved for line graphs by Reed and Seymour. We extend their result to all quasi‐line graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 17–33, 2008     

On factors of 4‐connected claw‐free graphs*

We consider the existence of several different kinds of factors in 4‐connected claw‐free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4‐connected line graph is hamiltonian, i.e., has a connected 2‐factor. Conjecture 2 (Matthews and Sumner): Every 4‐connected claw‐free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass‐free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 125–136, 2001     

On the hamiltonicity of line graphs of locally finite, 6‐edge‐connected graphs

The topological approach to the study of infinite graphs of Diestel and KÜhn has enabled several results on Hamilton cycles in finite graphs to be extended to locally finite graphs. We consider the result that the line graph of a finite 4‐edge‐connected graph is hamiltonian. We prove a weaker version of this result for infinite graphs: The line graph of locally finite, 6‐edge‐connected graph with a finite number of ends, each of which is thin, is hamiltonian.     

Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

In this paper, we show that if G is a 3‐edge‐connected graph with and , then either G has an Eulerian subgraph H such that , or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3‐edge‐connected planar graph, then for any , G has an Eulerian subgraph H such that . As an application, we obtain a new result on Hamiltonian line graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 308–319, 2003     

Symmetric Hamilton cycle decompositions of complete graphs minus a 1‐factor

Let n≥2 be an integer. The complete graph K_n with a 1‐factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that K_n−F has a decomposition into Hamilton cycles which are symmetric with respect to the 1‐factor F if and only if n≡2, 4 mod 8. We also show that the complete bipartite graph K_{n, n} has a symmetric Hamilton cycle decomposition if and only if n is even, and that if F is a 1‐factor of K_{n, n}, then K_{n, n}−F has a symmetric Hamilton cycle decomposition if and only if n is odd. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:1‐15, 2010     

Long cycles in 3‐connected graphs in orientable surfaces

In this article, we apply a cutting theorem of Thomassen to show that there is a function f: N → N such that if G is a 3‐connected graph on n vertices which can be embedded in the orientable surface of genus g with face‐width at least f(g), then G contains a cycle of length at least cn, where c is a constant not dependent on g. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 69–84, 2002     

On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors

Given two 2‐regular graphs F₁ and F₂, both of order n, the Hamilton‐Waterloo Problem for F₁ and F₂ asks for a factorization of the complete graph into α₁ copies of F₁, α₂ copies of F₂, and a 1‐factor if n is even, for all nonnegative integers α₁ and α₂ satisfying . We settle the Hamilton‐Waterloo Problem for all bipartite 2‐regular graphs F₁ and F₂ where F₁ can be obtained from F₂ by replacing each cycle with a bipartite 2‐regular graph of the same order.     

Spanning even subgraphs of 3‐edge‐connected graphs

By Petersen's theorem, a bridgeless cubic graph has a 2‐factor. H. Fleischner extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has a spanning even subgraph. Our main result is that, under the stronger hypothesis of 3‐edge‐connectivity, we can find a spanning even subgraph in which every component has at least five vertices. We show that this is in some sense best possible by constructing an infinite family of 3‐edge‐connected graphs in which every spanning even subgraph has a 5‐cycle as a component. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 37–47, 2009     

The Hamilton–Waterloo Problem for ‐Factors and ‐Factors

The Hamilton–Waterloo problem asks for a 2‐factorization of (for v odd) or minus a 1‐factor (for v even) into ‐factors and ‐factors. We completely solve the Hamilton–Waterloo problem in the case of C₃‐factors and ‐factors for .     

Hourglasses and Hamilton cycles in 4‐connected claw‐free graphs

We show that if G is a 4‐connected claw‐free graph in which every induced hourglass subgraph S contains two non‐adjacent vertices with a common neighbor outside S, then G is hamiltonian. This extends the fact that 4‐connected claw‐free, hourglass‐free graphs are hamiltonian, thus proving a broader special case of a conjecture by Matthews and Sumner. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 267–276, 2005     

An approximate version of Hadwiger's conjecture for claw‐free graphs

Hadwiger's conjecture states that every graph with chromatic number χ has a clique minor of size χ. In this paper we prove a weakened version of this conjecture for the class of claw‐free graphs (graphs that do not have a vertex with three pairwise nonadjacent neighbors). Our main result is that a claw‐free graph with chromatic number χ has a clique minor of size $\lceil\frac{2}{3}\chi\rceil$. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 259–278, 2010     

2‐connected 7‐coverings of 3‐connected graphs on surfaces

An m‐covering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3‐connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2‐connected 7‐covering with at most 6k ? 12 vertices of degree 7. We also construct, for every surface F² with Euler genus k ≥ 2, a 3‐connected graph G on F² with arbitrarily large representativity each of whose 2‐connected 7‐coverings contains at least 6k ? 12 vertices of degree 7. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 26–36, 2003     

Maximal K3's and Hamiltonicity of 4‐connected claw‐free graphs

Let cl(G) denote Ryjá?ek's closure of a claw‐free graph G. In this article, we prove the following result. Let G be a 4‐connected claw‐free graph. Assume that G[N_G(T)] is cyclically 3‐connected if T is a maximal K₃ in G which is also maximal in cl(G). Then G is hamiltonian. This result is a common generalization of Kaiser et al.'s theorem [J Graph Theory 48(4) (2005), 267–276] and Pfender's theorem [J Graph Theory 49(4) (2005), 262–272]. © 2011 Wiley Periodicals, Inc. J Graph Theory     

A note on 3‐partite graphs without 4‐cycles

Let $C_4$ be a cycle of order 4. Write $ex\left(n,n,n,C_4\right)$ for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has $n$ vertices that have no subgraph isomorphic to $C_4$. In this paper, we show that $ex\left(n,n,n,C_4\right)\ge \frac{3}{2}n\left(p+1\right)$, where $n=\frac{p\left(p?1\right)}{2}$ and $p$ is a prime number. Note that $ex\left(n,n,n,C_4\right)\le \left(\frac{3\sqrt{2}}{2}+o\left(1\right)\right)n^{\frac{3}{2}}$ from Tait and Timmons's works. Since for every integer $m$, one can find a prime $p$ such that $m\le p\le \left(1+o\left(1\right)\right)m$, we obtain that $\underset{n\to \infty }{\lim }\frac{ex\left(n,n,n,C_4\right)}{\frac{3\sqrt{2}}{2}{n}^{\frac{3}{2}}}=1$.     

Relative length of longest paths and cycles in 3‐connected graphs

For a graph G, let p(G) denote the order of a longest path in G and c(G) the order of a longest cycle in G, respectively. We show that if G is a 3‐connected graph of order n such that for every independent set {x₁, x₂, x₃, x₄}, then G satisfies c(G) ≥ p(G) ? 1. Using this result, we give several lower bounds to the circumference of a 3‐connected graph. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 137–156, 2001