Maximal K3's and Hamiltonicity of 4-connected claw-free graphs

Let cl(G) denote the Ryjáček’s closure of a claw-free graph G. In this paper, 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 (2005), no. 4, 267-276] and Pfender’s theorem [J. Graph Theory 49 (2005), no. 4, 262-272].