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...

A graph is claw-free if it does not contain K I ,3 as an induced subgraph. A graph is KI,r-free if it does not contain KI,r as an induced subgraph. In this paper, we find bounds on the minimum number of edges needed to ensure a KI,.,.-free contains k vertex disjoint cycles. The bound on claw-free graphs is sharp.

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.

To the American lobster (Homarus americanus) fishery in the Northeast US and Atlantic Canada, 2012 may have gone down as the year that drove climate change home. An ocean heat wave covered a large swath of the North Atlantic and broke ocean temperature records. It also triggered biological effects that upset the relatively harmonious economics of Canada’s winter and New England’s summer lobster...

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

Shelter dwelling and gregariousness are behavioral strategies used by benthic marine organisms to reduce the risk of predator-induced mortality. For spiny lobsters, the relative and synergistic success of shelter dwelling and aggregation in reducing mortality may vary with the perceived risk of predation as well as the availability and size distribution of conspecifics and shelters. We tested h...

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...