Forbidden Subgraphs for Hamiltonicity of 3-Connected Claw-Free Graphs

In this paper, we consider forbidden subgraphs for hamiltonicity of 3-connected claw-free graphs. Let Zi be the graph obtaind from a triangle by attaching a path of length i to one of its vertices, and let Q∗ be the graph obtained from the Petersen graph by adding one pendant edge to each vertex. Lai et al. [J. Graph Theory 64 (2010), no. 1, 1-11] conjectured that every 3-connected {K1,3, Z9}-free graph G is hamiltonian unless G is the line graph of Q ∗. It is shown in this paper that this conjecture is true. Moreover, we investigate the set of graphs A3 which satisfies that every 3-connected {K1,3, A}-free graph of sufficiently large order is hamiltonian if and only if A is a member of A3. We prove that, if G ∈ A3, then G is a line graph, |V (G)| ≤ 16, the diameter of G is at most 10, the circumference of G is at most 3, ∆(G) ≤ 3, and there are at most 5 triangles in G.