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

  • Junfeng Du
  • Liming XiongEmail author
Original Paper


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 \(``i+j\le 10\)”.


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



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).


  1. 1.
    Bedrossian, P.: Forbidden subgraph and minimum degree conditions for hamiltonicity. Memphis State University, USA, Thesis (1991)Google Scholar
  2. 2.
    Bondy, J.A., Murty, U.S.R.: Graph theory with applications. Macmillan, London and Elsevier, New York (1976)Google Scholar
  3. 3.
    Brousek, J., Ryjáček, Z., Favaron, O.: Forbidden subgraphs, hamiltonicity and closure in claw-free graphs. Discrte Math. 196, 29–50 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brousek, J., Ryjáček, Z., Schiermeyer, I.: Forbidden subgraphs, stability and hamiltonicity. Discrte Math. 197(198), 143–155 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Faudree, R.J., Gould, R.J.: Characterizing forbidden pairs for Hamiltonian properties. Discrete Math. 173, 45–60 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fujisawa, J.: Forbidden subgraphs for hamiltonicity of 3-connected claw-free graphs. J. Graph Theory 73, 146–160 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Harary, F., Nash-Williams, C.S.T.J.A.: On eulerian and hamiltonian graphs and line graphs. Can. Math. Bull. 8, 701–709 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hu, Z., Lin, H.: Two forbidden subgraph pairs for hamiltonicity of 3-connected graphs. Graphs Combin. 29, 1755–1775 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lai, H.-J., Xiong, L., Yan, H., Yan, J.: Every 3-connected claw-free \(Z_{8}\)-free graph is hamiltonian. J. Graph Theory 64, 1–11 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, B., Vrána, P.: Forbidden pairs of disconnected graphs implying hamiltonicity. J. Graph Theory 84, 249–261 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Luczak, T., Pfender, F.: Claw-free 3-connected \(P_{11}\)-free graphs are hamiltonian. J. Graph Theory 47, 111–121 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ryjáček, Z.: On a closure concept in claw-Free graphs. J. Combin. Theory Ser. B 70, 217–224 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ryjáček, Z., Vrána, P., Xiong, L.: Hamiltonian properties of 3-connected \(\{\)claw, hourglass\(\}\)-free graphs. Discrete Math. 341, 1806–1815 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Xiong, W., Lai, H.-J., Ma, X.L., Wang, K., Zhang, M.: Hamilton cycles in 3-connected claw-free and net-free graphs. Discrete Math. 313, 784–795 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations