For most graphs H,most H‐free graphs have a linear homogeneous set

Erd?s and Hajnal conjectured that for every graph H there is a constant such that every graph G that does not have H as an induced subgraph contains a clique or a stable set of order . The conjecture would be false if we set ; however, in an asymptotic setting, we obtain this strengthened form of Erd?s and Hajnal's conjecture for almost every graph H, and in particular for a large class of graphs H defined by variants of the colouring number. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 343–361, 2014     

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 complexity of reconstructing H‐free graphs from their Star Systems

In the Star System problem we are given a set system and asked whether it is realizable by the multi‐set of closed neighborhoods of some graph, i.e. given subsets S₁, S₂, …, S_n of an n‐element set V does there exist a graph G = (V, E) with {N[v]: v∈V} = {S₁, S₂, …, S_n}? For a fixed graph H the H‐free Star System problem is a variant of the Star System problem where it is asked whether a given set system is realizable by closed neighborhoods of a graph containing no H as an induced subgraph. We study the computational complexity of the H‐free Star System problem. We prove that when H is a path or a cycle on at most four vertices the problem is polynomial time solvable. In complement to this result, we show that if H belongs to a certain large class of graphs the H‐free Star System problem is NP‐complete. In particular, the problem is NP‐complete when H is either a cycle or a path on at least five vertices. This yields a complete dichotomy for paths and cycles. Copyright © 2010 John Wiley & Sons, Ltd. 68:113‐124, 2011     

Construction of uniquely H‐colorable graphs

We shall prove that for any graph H that is a core, if χ(G) is large enough, then H × G is uniquely H‐colorable. We also give a new construction of triangle free graphs, which are uniquely n‐colorable. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 1–6, 1999     

Claw‐free circular‐perfect graphs

The circular chromatic number of a graph is a well‐studied refinement of the chromatic number. Circular‐perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This article studies claw‐free circular‐perfect graphs. First, we prove that if G is a connected claw‐free circular‐perfect graph with χ(G)>ω(G), then min{α(G), ω(G)}=2. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw‐free circular‐perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs G have min{α(G), ω(G)}=2. In contrast to this result, it is shown in Z. Pan and X. Zhu [European J Combin 29(4) (2008), 1055–1063] that minimal circular‐imperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. In this article, we prove that claw‐free minimal circular‐imperfect graphs G have min{α(G), ω(G)}≤3. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 163–172, 2010     

Clique‐inverse graphs of K3‐free and K4‐free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique‐inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique‐inverse graphs of K₃‐free and K₄‐free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000     

On triangle‐free random graphs

We show that for every k≥1 and δ>0 there exists a constant c>0 such that, with probability tending to 1 as n→∞, a graph chosen uniformly at random among all triangle‐free graphs with n vertices and M≥cn^3/2 edges can be made bipartite by deleting ⌊δM⌋ edges. As an immediate consequence of this fact we infer that if M/n^3/2→∞ but M/n²→0, then the probability that a random graph G(n, M) contains no triangles decreases as 2^−(1+o(1))M. We also discuss possible generalizations of the above results. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 260–276, 2000     

Critical graphs for subpancyclicity of 3‐connected claw‐free graphs

Let be the family of graphs G such that all sufficiently large k ‐connected claw‐free graphs which contain no induced copies of G are subpancyclic. We show that for every k≥3 the family is infinite and make the first step toward the complete characterization of the family . © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 263–278, 2009     

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     

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     

Scale‐free graphs of increasing degree

We study a scale‐free random graph process in which the number of edges added at each step increases. This differs from the standard model in which a fixed number, m, of edges are added at each step. Let f(t) be the number of edges added at step t. In the standard scale‐free model, f(t) = m constant, whereas in this paper we consider f(t) = [t^c],c > 0. Such a graph process, in which the number of edges grows non‐linearly with the number of vertices is said to have accelerating growth. We analyze both an undirected and a directed process. The power law of the degree sequence of these processes exhibits widely differing behavior. For the undirected process, the terminal vertex of each edge is chosen by preferential attachment based on vertex degree. When f(t) = m constant, this is the standard scale‐free model, and the power law of the degree sequence is 3. When f(t) = [t^c],c < 1, the degree sequence of the process exhibits a power law with parameter x = (3 ? c)/(1 ? c). As c → 0, x → 3, which gives a value of x = 3, as in standard scale‐free model. Thus no more slowly growing monotone function f(t) alters the power law of this model away from x = 3. When c = 1, so that f(t) = t, the expected degree of all vertices is t, the vertex degree is concentrated, and the degree sequence does not have a power law. For the directed process, the terminal vertex is chosen proportional to in‐degree plus an additive constant, to allow the selection of vertices of in‐degree zero. For this process when f(t) = m is constant, the power law of the degree sequence is x = 2 + 1/m. When f(t) = [t^c], c > 0, the power law becomes x = 1 + 1/(1 + c), which naturally extends the power law to [1,2]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 396–421, 2011     

Dynamic choosability of triangle‐free graphs and sparse random graphs

Ther‐dynamic choosability of a graph G, written , is the least k such that whenever each vertex is assigned a list of at least k colors a proper coloring can be chosen from the lists so that every vertex v has at least neighbors of distinct colors. Let ch(G) denote the choice number of G. In this article, we prove when is bounded. We also show that there exists a constant C such that the random graph with almost surely satisfies . Also if G is a triangle‐free regular graph, then we have .     

Claw‐free 3‐connected P11‐free graphs are hamiltonian

We show that every 3‐connected claw‐free graph which contains no induced copy of P₁₁ is hamiltonian. Since there exist non‐hamiltonian 3‐connected claw‐free graphs without induced copies of P₁₂ this result is, in a way, best possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 111–121, 2004     

Complexity of clique‐coloring odd‐hole‐free graphs

In this paper we investigate the problem of clique‐coloring, which consists in coloring the vertices of a graph in such a way that no monochromatic maximal clique appears, and we focus on odd‐hole‐free graphs. On the one hand we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, but on the other hand it is NP‐hard to decide if they are 2‐clique‐colorable, and we do not know if there exists any bound k₀ such that they are all k₀ ‐clique‐colorable. First we will prove that (odd hole, codiamond)‐free graphs are 2‐clique‐colorable. Then we will demonstrate that the complexity of 2‐clique‐coloring odd‐hole‐free graphs is actually Σ₂ P‐complete. Finally we will study the complexity of deciding whether or not a graph and all its subgraphs are 2‐clique‐colorable. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 139–156, 2009     

Exhaustive generation of k‐critical ‐free graphs

We describe an algorithm for generating all k‐critical ‐free graphs, based on a method of Hoàng et al. (A graph G is k‐critical H‐free if G is H‐free, k‐chromatic, and every H‐free proper subgraph of G is ‐colorable). Using this algorithm, we prove that there are only finitely many 4‐critical ‐free graphs, for both and . We also show that there are only finitely many 4‐critical ‐free graphs. For each of these cases we also give the complete lists of critical graphs and vertex‐critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the 3‐colorability problem in the respective classes. In addition, we prove a number of characterizations for 4‐critical H‐free graphs when H is disconnected. Moreover, we prove that for every t, the class of 4‐critical planar ‐free graphs is finite. We also determine all 52 4‐critical planar P₇‐free graphs. We also prove that every P₁₁‐free graph of girth at least five is 3‐colorable, and show that this is best possible by determining the smallest 4‐chromatic P₁₂‐free graph of girth at least five. Moreover, we show that every P₁₄‐free graph of girth at least six and every P₁₇‐free graph of girth at least seven is 3‐colorable. This strengthens results of Golovach et al.     

Diameter in ultra‐small scale‐free random graphs

It is well known that many random graphs with infinite variance degrees are ultra‐small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k^{?(τ ? 1)} with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to and , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order precisely when the minimal forward degree d_fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.     

Upper total domination in claw‐free graphs

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S (other than itself). The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γ_t(G). We establish bounds on Γ_t(G) for claw‐free graphs G in terms of the number n of vertices and the minimum degree δ of G. We show that if if , and if δ ≥ 5. The extremal graphs are characterized. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 148–158, 2003     

Rooted induced trees in triangle‐free graphs

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceilFor a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceil$, improving a recent result by Fox, Loh and Sudakov. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 206–209, 2010