Vertex Pancyclicity of Quadrangularly Connected Claw-free Graphs

Abstract

A graph $$G$$G is quadrangularly connected if for every pair of edges $$e_1$$e1 and $$e_2$$e2 in $$E(G)$$E(G), $$G$$G has a sequence of $$l$$l-cycles ($$3\le l\le 4$$3≤l≤4) $$C_1, C_2,\ldots ,C_r$$C1,C2,�,Cr such that $$e_1\in E(C_1),$$e1�E(C1),$$e_2\in E(C_r)$$e2�E(Cr) and $$E(C_i)\cap E(C_{i+1})\ne \emptyset $$E(Ci)�E(Ci+1)�� for $$i=1,2,\ldots ,r-1.$$i=1,2,�,r-1. In this paper, we show that if $$G$$G is a quadrangularly connected claw-free graph with $$\delta (G)\ge 5$$�(G)�5, which does not contain an induced subgraph $$H$$H isomorphic to either $$G_1$$G1 or $$G_2$$G2 (where $$G_1$$G1, $$G_2$$G2 are specified graphs on 8 vertices) such that the neighborhood in $$G$$G of every vertex of degree 4 in $$H$$H is disconnected, then $$G$$G is vertex pancyclic.