Cycles through particular subgraphs of claw-free graphs

Let G be a 2-connected claw-free graph on n vertices, and let H be a subgraph of G. We prove that G has a cycle containing all vertices of H whenever a3(H) 5 K(G), where a 3 ( H ) denotes the maximum number of vertices of H that are pairwise at distance at least three in G, and K(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 K(G) + 1 pairwise nonadjacent vertices is at least n K(G).