Quadrangularly connected claw-free graphs

A graph G is quadrangularly connected if for every pair of edges image and image in image, G has a sequence of l-cycles image image such that image and image and image for image. 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 image or image such that image of every vertex x of degree 4 in H is disconnected is hamiltonian, which implies a result by Z. Ryjác˘ek [Hamiltonian circuits in image-locally connected image-free graphs, J. Graph Theory 14 (1990) 321–331] and other known results.