CYCLES THROUGH PARTICULAR SUBGRAPHS OF CLAW-FREE GRAPHS

Let G be a 2-connected claw-free graph on n vertices, and let Hbe a subgraph of G. We prove that G has a cycle containing all vertices of H whenever alpha(3)(H) less than or equal to kappa(G), where alpha(3)(H) denotes the maximum number of vertices of H that are pairwise at distance at least three in G, and kappa(G) denotes the connectivity of G. This result is an analog of a result from the thesis of Fournier, and generalizes the result of Zhang that G is hamiltonian if the degree sum of any kappa(G) + 1 pairwise nonadjacent vertices is at least n - kappa(G). (C) 1995 John Wiley & Sons, Inc.