Quadrangularly connected claw-free graphs

AbstractA graph G is quadrangularly connected if for every pair of edges e1 and e2 in E(G), G has a sequence of l-cycles (3≤l≤4) C1,C2,…,Cr such that e1∈E(C1) and e2∈E(Cr) and E(Ci)∩E(Ci+1)≠∅ for i=1,2,…,r-1. In this paper, we show that every quadrangularly connected claw-free graph without vertices of degree 1, which does not contain an induced subgraph H isomorphic to either G1 or G2 such that N1(x,G) of every vertex x of degree 4 in H is disconnected is hamiltonian, which implies a result by Z. Ryjác˘ek [Hamiltonian circuits in N2-locally connected K1,3-free graphs, J. Graph Theory 14 (1990) 321–331] and other known results