The Local Structure of Claw-Free Graphs Without Induced Generalized Bulls

In this paper, we show the following: Let G be a connected claw-free graph such that G has a connected induced subgraph H that has a pair of vertices {v1,v2} of degree one in H whose distance is d+2in H. Then H has an induced subgraph F, which is isomorphic to Bi, j, with {v1, v2}. V( F) and i + j = d + 1, with a well-defined exception. Here Bi, j denotes the graph obtained by attaching two vertex-disjoint paths of lengths i, j = 1 to a triangle. We also use the result above to strengthen the results in Xiong et al. ( Discrete Math 313: 784-795, 2013) in two cases, when i + j = 9, and when the graph is 0-free. Here 0 is the simple graph with degree sequence 4, 2, 2, 2, 2. Let i, j > 0 be integers such that i + j = 9. Then every 3-connected {K1,3, Bi, j}-free graph G is hamiltonian, and every 3-connected {K1,3, 0, B2i,2 j}-free graph G is hamiltonian. The two results above are all sharp in the sense that the condition " i + j = 9" couldn't be replaced by " i + j <= 10".