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 \(n \leq 3\delta+2\). 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.