A connected graph G with chromatic number t is double-critical if G − x − y is (t − 2)-colorable for each edge xy ∈ E(G). The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erdős and Lovász from 1966, which is referred to as the Double-critical Graph Conjecture, states that there are no other double-critical graphs, i.e., if a graph G with c...

Let G be any graph and let c(G) denote the circumference of G. We conjecture that for every pair c1, c2 of positive integers satisfying c1 + c2 = c(G), the vertex set of G admits a partition into two sets V1 and V2, such that Vi induces a graph of circumference at most ci , i = 1, 2. We establish various results in support of the conjecture; e.g. it is observed that planar graphs, claw-free gra...

We survey results and open problems in hamiltonian graph theory centred around three themes: regular graphs, t-tough graphs, and claw-free graphs.

We determine the maximum number of edges that a claw-free graph can have, when its maximum degree and matching number are bounded. This is a famous problem that has been studied on general graphs, and for which there is a tight bound. The graphs achieving this bound contain in most cases an induced copy of K1,3, the claw, which motivates studying the question on claw-free graphs. Note that on g...

A graph G is Nm-locally connected if for every vertex v in G, the vertices not equal to v and with distance at most m to v induce a connected subgraph in G. We show that both connected N2-locally connected claw-free graph and 3-edge-connected N3-locally connected claw-free graph have connected even [2, 4]-factors, which settle a conjecture by Li in [6].

It is proven that if G is a 3-connected claw-free graph which is also Z3-free (where Z3 is a triangle with a path of length 3 attached), P6-free (where P6 is a path with 6 vertices) or H1-free (where H1 consists of two disjoint triangles connected by an edge), then G is hamiltonian-connected. Also, examples will be described that determine a finite family of graphs L such that if a 3-connected ...

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [9,14] and claw-free graphs [8,9]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [10]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far,...

Graphs with circular symmetry, called webs, are relevant for describing the stable set polytopes of two larger graph classes, quasi-line graphs [6,10] and claw-free graphs [5,6]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a longstanding problem [7]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it i...

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [8,12] and claw-free graphs [7,8]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] v...

We show that deciding whether a given graph G of size m has a unique perfect matching as well as finding that matching, if it exists, can be done in time O(m) if G is either a cograph, or a split graph, or an interval graph, or claw-free. Furthermore, we provide a constructive characterization of the claw-free graphs with a unique perfect matching.