A Hamilton circle in an infinite graph is a homeomorphic copy of the unit circle S1 that contains all vertices and all ends precisely once. We prove that every connected, locally connected, locally finite, claw-free graph has such a Hamilton circle, extending a result of Oberly and Sumner to infinite graphs. Furthermore, we show that such graphs are Hamilton-connected if and only if they are 3-...

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

We study the stability of some classes of graphs deened in terms of forbidden sub-graphs under the closure operation introduced by the second author. Using these results, we prove that every 2-connected claw-free and P 7-free, or claw-free and Z 4-free, or claw-free and eiiel-free graph is either hamiltonian or belongs to a certain class of exceptions (all of them having connectivity 2).

A graph G is a minimal claw-free graph (m.c.f. graph) if it contains no K1,3 (claw) as an induced subgraph and if, for each edge e of G, G − e contains an induced claw. We investigate properties of m.c.f. graphs, establish sharp bounds on their orders and the degrees of their vertices, and characterize graphs which have m.c.f. line graphs. MSC 2000: 05C75, 05C07

We prove that if G is a quasi-line graph with ∆(G) > ω(G) and ∆(G) ≥ 69, then χOL(G) ≤ ∆(G) − 1. Together with our previous work, this implies that if G is a claw-free graph with ∆(G) > ω(G) and ∆(G) ≥ 69, then χl(G) ≤ ∆(G)− 1.

Let G be a claw-free graph with order n and minimum degree δ. We improve results of Faudree et al. and Gould & Jacobson, and solve two open problems by proving the following two results. If δ = 4, then G has a 2-factor with at most (5n− 14)/18 components, unless G belongs to a finite class of exceptional graphs. If δ ≥ 5, then G has a 2-factor with at most (n− 3)/(δ − 1) components, unless G is...

This paper provides a complete characterization of all the rank facets of the stable set polytope STAB(G) associated with a claw-free graph G. In particular, it is shown that a claw-free graph G produces a rank facet of STAB(G) if and only if it can be obtained by means of two simple lifting procedures from three basic classes of graphs: (i) cliques, (ii) line graphs of minimal 2-connected hypo...

Bilateral asymmetry of the paired claws of the lobster Homarus americanus is determined during the fourth and fifth juvenile stages by differential reflex activity; the side with the greater activity becomes the crusher while the contralateral side becomes the cutter. Juvenile lobsters reared during this critical period with a substratum that could not be grasped or with reduced input from pred...

To help understand the process of carpel morphogenesis, the roles of three carpel development genes have been partitioned genetically. Mutants of CRABS CLAW cause the gynoecium to develop into a wider but shorter structure, and the two carpels are unfused at the apex. Mutants of a second gene, SPATULA, show reduced growth of the style, stigma, and septum, and the transmitting tract is absent. D...

We prove that every (claw, net)-free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present a linear time algorithm which, for a given (claw, net)-free graph, 3nds either a dominating pair or an induced doubly dominating cycle. We show also how one can use structural properties of (claw, net)-free graphs to solve e4ciently the domination, indep...