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

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## Abstract

In this paper, we show the following: Let The two results above are all sharp in the sense that the condition “\(i+j\le 9\)” couldn’t be replaced by \(``i+j\le 10\)”.

*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.

## Keywords

Forbidden subgraph Claw-free Closure 3-Connected graph Generalized bull## Notes

### Acknowledgements

The work is supported by the Natural Science Funds of China (Nos. 11871099, 11671037) and by the Nature Science Foundation from Qinghai Province (No. 2018-ZJ-717).

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