Quadrangularly connected claw-free graphs

A graph G is quadrangularly connected   if for every pair of edges e1e1 and e2e2 in E(G)E(G), G has a sequence of l  -cycles (3≤l≤4)(3≤l≤4)C1,C2,…,CrC1,C2,…,Cr such that e1∈E(C1)e1∈E(C1) and e2∈E(Cr)e2∈E(Cr) and E(Ci)∩E(Ci+1)≠∅E(Ci)∩E(Ci+1)≠∅ for i=1,2,…,r-1i=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 G1G1 or G2G2 such that N1(x,G)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 N2N2-locally connected K1,3K1,3-free graphs, J. Graph Theory 14 (1990) 321–331] and other known results.