A 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,…,C_r such that e₁E(C₁) and e₂E(C_r) and E(C_i)∩E(C_i+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 K_1,3-free graphs, J. Graph Theory 14 (1990) 321–331] and other known results.