Let cl(G) denote Ryjáček's closure of a claw-free graph G. In this article, we prove the following result. Let G be a 4-connected claw-free graph. Assume that G[NG(T)] is cyclically 3-connected if T is a maximal K3 in G which is also maximal in cl(G). Then G is hamiltonian. This result is a common generalization of Kaiser et al.'s theorem [J Graph Theory 48(4) (2005), 267-276] and Pfender's theorem [J Graph Theory 49(4) (2005), 262-272].
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