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

  • Junfeng Du
  • Liming XiongEmail author
Original Paper
  • 13 Downloads

Abstract

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 \(``i+j\le 10\)”.

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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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingPeople’s Republic of China
  2. 2.School of Mathematics and Statistics, Beijing Key Laboratory on MCAACIBeijing Institute of TechnologyBeijingPeople’s Republic of China

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