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 \(\{v_{1}, v_{2}\}\) of degree one in H whose distance is \(d + 2\) in H. Then H has an induced subgraph F, which is isomorphic to \(B_{i,j}\), with \(\{v_{1}, v_{2}\} \subseteq V(F)\) and \(i+j=d+1\), with a well-defined exception. Here \(B_{i, j}\) denotes the graph obtained by attaching two vertex-disjoint paths of lengths \(i, j \ge 1\) 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 \(i + j \le 9\), and when the graph is \(\Gamma _{0}\)-free. Here \(\Gamma _{0}\) is the simple graph with degree sequence 4, 2, 2, 2, 2. Let \(i, j > 0\) be integers such that \(i + j \le 9\). Then every 3-connected \(\{K_{1,3}, B_{i, j}\}\)-free graph G is hamiltonian, and every 3-connected \(\{K_{1,3}, \Gamma _{0}, B_{2i, 2j}\}\)-free graph G is hamiltonian. The two results above are all sharp in the sense that the condition “\(i+j\le 9\)” couldn’t be replaced by \(``i+j\le 10\)”.