Quadrangularly connected claw-free graphs

A graph G is quadrangularly connected if for every p. air of edges e(1) and e(2) in E(G), G has a sequence of l-cycles (3 <= 1 <= 4) C-1, C-2,..., C-r such that e(1) is an element of E(C-1) and e(2) is an element of E(C-r) and E(C-i) boolean AND E(Ci+1) not equal 0 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 G(1) or G(2) such that N-1 (x, G) of every vertex x of degree 4 in H is disconnected is hamiltonian, which implies a result by Z. Ryjacek [Hamiltonian circuits in N-2-locally connected K-1,K-3-free graphs, J. Graph Theory 14 (1990) 321-331] and other known results. (c) 2006 Elsevier B.V. All rights reserved.