Claw-free 3-connected P11-free graphs are hamiltonian

We show that every 3-connected claw-free graph which contains no induced copy of P11 is hamiltonian. Since there exist non-hamiltonian 3-connected claw-free graphs without induced copies of P12 this result is, in a way, best possible. 1. Statement of the main result A graph G is {H1, H2, . . . Hk}-free if G contains no induced subgraphs isomorphic to any of the graphs Hi, i = 1, 2, . . . , k. A graph without induced copies of K1,3 is called claw-free, and a graph containing no copies of K3 is triangle-free. Broersma and Veldman [3] showed the following theorem. (Here and below Pk denotes the path on k vertices.) Theorem 1. If G is a 2-connected {K1,3, P6}-free graph, then G is hamiltonian. Bedrossian [1] characterized all pairs of forbidden subgraphs X, Y , such that every 2-connected {X, Y }-free graph is hamiltonian. Later, Faudree and Gould [6] extended that list under the extra condition that the graph has at least ten vertices. In this paper we study 3-connected graphs and show the following result analogous to Theorem 1. Theorem 2. Every 3-connected {K1,3, P11}-free graph is hamiltonian. This extends a result from Brousek et al. [5], who showed as a corollary of a result about 2-connected claw-free graphs that every 3connected {K1,3, P7}-free graph is hamiltonian. Furthermore, in the last section of the paper, we give an example of a 3-connected {K1,3, P12}-free graph which is not hamiltonian. 1991 Mathematics Subject Classification. 05C45.