Vertex Pancyclicity of Quadrangularly Connected Claw-free Graphs

Abstract

A graph $G$ is quadrangularly connected if for every pair of edges $e_1$ and $e_2$ in $E(G)$, $G$ has a sequence of $l$-cycles ($3\le l\le 4$) $C_1, C_2,\ldots ,C_r$ such that $e_1\in E(C_1),$ $e_2\in E(C_r)$ and $E(C_i)\cap E(C_{i+1})\ne \emptyset$ for $i=1,2,\ldots ,r-1.$ In this paper, we show that if $G$ is a quadrangularly connected claw-free graph with $\delta (G)\ge 5$, which does not contain an induced subgraph $H$ isomorphic to either $G_1$ or $G_2$ (where $G_1$, $G_2$ are specified graphs on 8 vertices) such that the neighborhood in $G$ of every vertex of degree 4 in $H$ is disconnected, then $G$ is vertex pancyclic.