Generalized Pancyclic Properties in Claw-free Graphs

Abstract

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