A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a minimum clique-transversal and a maximum clique-independent set of G, respectively. A graph G is clique-perfect if these two numbers are equal ...

In this paper, we consider forbidden subgraphs for hamiltonicity of 3-connected claw-free graphs. Let Zi be the graph obtaind from a triangle by attaching a path of length i to one of its vertices, and let Q∗ be the graph obtained from the Petersen graph by adding one pendant edge to each vertex. Lai et al. [J. Graph Theory 64 (2010), no. 1, 1-11] conjectured that every 3-connected {K1,3, Z9}-f...

A graph is h-perfect if its stable set polytope can be completely described by nonnegativity, clique and odd-hole constraints. It is t-perfect if it furthermore has no clique of size 4. For every graph G and every c ∈ Z (G) + , the weighted chromatic number of (G, c) is the minimum cardinality of a multi-set F of stable sets of G such that every v ∈ V (G) belongs to at least cv members of F . W...

An undirected graph is commonly represented as a set of vertices and a set of doubletons of vertices; but one can also represent vertices by finite sets so as to ensure that membership mimics, over those sets, the edge relation of the graph. This alternative modeling, applied to connected claw-free graphs, recently gave crucial clues for obtaining simpler proofs of some of their properties (e.g...

Wesurvey results and open problems in hamiltonian graph theory centered around two conjectures of the 1980s that are still open: every 4-connected claw-free graph (line graph) is hamiltonian. These conjectures have lead to awealth of interesting concepts, techniques, results and equivalent conjectures.

Ryjáček introduced a closure concept in claw-free graphs based on local completion at a locally connected vertex. He showed that the closure of a graph is the line graph of a triangle-free graph. Brousek and Holub gave an analogous closure concept of claw-free graphs, called the edge-closure, based on local completion at a locally connected edge. In this paper, it is shown that the edge-closure...

Consider the triangle-free graph process, which starts from the empty graph on n vertices and in every step an edge is added that is chosen uniformly at random from all non-edges that do not form a triangle with the existing edges. We will show that there exists a constant c such that asymptotically almost surely no copy of any fixed finite triangle-free graph on k vertices with at least ck edg...

A graph G is k-Hamilton-connected (k-hamiltonian) if G−X is Hamilton-connected (hamiltonian) for every setX ⊂ V (G) with |X| = k. In the paper, we prove that (i) every 5-connected claw-free graph with minimum degree at least 6 is 1Hamilton-connected, (ii) every 4-connected claw-free hourglass-free graph is 1-Hamilton-connected. As a byproduct, we also show that every 5-connected line graph with...

Hadwiger’s Conjecture [7] states that every Kt+1-minor-free graph is t-colourable. It is widely considered to be one of the most important conjectures in graph theory; see [21] for a survey. If every Kt+1-minor-free graph has minimum degree at most δ, then every Kt+1minor-free graph is (δ+1)-colourable by a minimum-degree-greedy algorithm. The purpose of this note is to prove a slightly better ...

A graph G is called Berge if neither G nor its complement contains a chordless cycle with an odd number of nodes. The famous Berge’s Strong Perfect Graph Conjecture asserts that every Berge graph is perfect. A chair is a graph with nodes {a, b, c, d, e} and edges {ab, bc, cd, eb}. We prove that a Berge graph with no induced chair (chair-free) is perfect or, equivalently, that the Strong Perfect...