Hamiltonian Type Properties in Claw-Free $P_5$-Free Graphs

Abstract

A graph G on n vertices is said to be (k, m)-pancyclic if every set of k vertices in G is contained in a cycle of length r for each integer r in the set $\{ m, m + 1, \ldots , n \}$. This property, which generalizes the notion of a vertex pancyclic graph, was defined by Faudree et al. in (Graphs Combin 20:291–310, 2004). The notion of (k, m)-pancyclicity provides one way to measure the prevalence of cycles in a graph. Broersma and Veldman showed in (Contemporary methods in graph theory, BI-Wiss.-Verlag, Mannheim, Wien, Zürich, pp 181–194, 1990) that any 2-connected claw-free $P_5$-free graph must be hamiltonian. In fact, every non-hamiltonian cycle in such a graph is either extendable or very dense. We show that any 2-connected claw-free $P_5$-free graph is (k, 3k)-pancyclic for each integer $k \ge 2$. We also show that such a graph is (1, 5)-pancyclic. Examples are provided which show that these results are best possible. Each example we provide represents an infinite family of graphs.