On Pancyclic Claw-Free Graphs

In this paper, we show that if \(G\) is a connected \(SN2\)-locally connected claw-free graph with \(\delta(G) \geq 3\), which does not contain an induced subgraph \(H\) isomorphic to either \(G_1\) or \(G_2\) such that \(N_1(x,G)\) of every vertex \(x\) of degree \(4\) in \(H\) is disconnected, then every \(N_2\)-locally connected vertex of \(G\) is contained in a cycle of all possible lengths and so \(G\) is pancyclic. Moreover, \(G\) is vertex pancyclic if \(G\) is \(N_2\)-locally connected.