Generalized Pancyclic Properties in Claw-free Graphs

Abstract

The property $$(k, m)$$(k,m)-pancyclicity, which generalizes the notion of a vertex pancyclic graph, was defined in Faudree et al. (Graphs Comb 20:291---310, 2004) Given integers $$k$$k and $$m$$m with $$k \le m \le n$$k≤m≤n, a graph $$G$$G of order $$n$$n is said to be $$(k, m)$$(k,m)-pancyclic if every set of $$k$$k vertices in $$G$$G is contained in a cycle of length $$r$$r, for each $$r \in \{ m, m + 1, \ldots , n \}$$r?{m,m+1,?,n}. Faudree et al. (Graphs Comb 20:291---310, 2004) established sharp Ore-type bounds which guarantee that a graph is $$(k, m)$$(k,m)-pancyclic for certain integers $$k$$k and $$m$$m. In particular, they proved that if $$\sigma _2(G) \ge n + 1$$?2(G)?n+1, then $$G$$G is $$(k, 2k)$$(k,2k)-pancyclic for each $$k \ge 2$$k?2. We show that if $$G$$G is claw-free and $$\sigma _2(G) \ge n$$?2(G)?n, then $$G$$G is $$(k, k + 3)$$(k,k+3)-pancyclic for each $$k \ge 1$$k?1. Other minimum degree sum conditions for nonadjacent vertices that imply a claw-free graph is $$(k, m)$$(k,m)-pancyclic are established, and examples are provided which show that these constraints are best possible.