Circumferences of $2$-Connected Claw-Free Graphs

Abstract

A known result due to Matthews and Sunner is that every $2$-connected claw-free graph on $n$ vertices contains a cycle of length at least $min\{2\delta +4,n\}$, and is Hamiltonian if $n\le 3\delta +2$. In this paper, we show that every $2$-connected claw-free graph on $n$ vertices which does not belong to one of three classes of exceptional graphs contains a cycle of length at least $min\{4\delta -2,n\}$, hereby generalizing several known results. Moreover, the bound $4\delta -2$ is almost best possible.