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Graphs and Combinatorics

, 25:427 | Cite as

Sharp Upper Bounds on the Minimum Number of Components of 2-factors in Claw-free Graphs

  • Hajo Broersma
  • Daniël Paulusma
  • Kiyoshi Yoshimoto
Article

Abstract

Let G be a claw-free graph with order n and minimum degree δ. We improve results of Faudree et al. and Gould & Jacobson, and solve two open problems by proving the following two results. If δ = 4, then G has a 2-factor with at most (5n − 14)/18 components, unless G belongs to a finite class of exceptional graphs. If δ ≥ 5, then G has a 2-factor with at most (n − 3)/(δ − 1) components, unless G is a complete graph. These bounds are best possible in the sense that we cannot replace 5/18 by a smaller quotient and we cannot replace δ − 1 by δ, respectively.

Keywords

Claw-free graph 2-factor minimum degree edge-degree 

References

  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Heidelberg (2007)Google Scholar
  2. 2.
    Choudum, S.A., Paulraj, M.S.: Regular factors in K 1,3-free graphs. J. Graph Theory 15, 259–265 (1991)Google Scholar
  3. 3.
    Diestel, R.: Graph Theory, 2nd ed., Graduate Texts in Mathematics, Vol. 173. Springer, Heidelberg (2000)Google Scholar
  4. 4.
    Egawa, Y., Ota, K.: Regular factors in K 1,n-free graphs. J. Graph Theory 15, 337–344 (1991)Google Scholar
  5. 5.
    Egawa, Y., Saito, A.: Private communicationsGoogle Scholar
  6. 6.
    Faudree, R.J., Favaron, O., Flandrin, E., Li, H., Liu, Z.: On 2-factors in claw-free graphs. Discrete Math. 206, 131–137 (1999)Google Scholar
  7. 7.
    Faudree, R., Flandrin, R., Ryjáček, Z.: Claw-free graphs—a survey. Discrete Math. 164, 87–147 (1997)Google Scholar
  8. 8.
    Fleischner, H.: Spanning Eulerian subgraphs, the splitting lemma, and Petersen’s theorem. Discrete Math. 101, 33–37 (1992)Google Scholar
  9. 9.
    Fronček, D., Ryjáček, Z., Skupień, Z.: On traceability and 2-factors in claw-free graphs. Discuss. Math. Graph Theory 24, 55–71 (2004)Google Scholar
  10. 10.
    Jackson, B., Yoshimoto, K.: Even subgraphs of bridgeless graphs and 2-factors of line graphs. Discrete Math. 307, 2775–2785 (2007)Google Scholar
  11. 11.
    Jackson, B., Yoshimoto, K.: Spanning even subgraphs of 3-edge-connected graphs. J. Graph Theory, (to appear)Google Scholar
  12. 12.
    Gould, R.J., Hynds, E.: A note on cycles in 2-factors of line graphs. Bull. ICA 26, 46–48 (1999)Google Scholar
  13. 13.
    Gould, R.J., Jacobson, M.S.: Two-factors with few cycles in claw-free graphs. Discrete Math. 231, 191–197 (2001)Google Scholar
  14. 14.
    Harary, F., Nash-Williams, C.St.J.A.: On eulerian and hamiltonian graphs and line graphs. Can. Math. Bull. 8, 701–710 (1965)Google Scholar
  15. 15.
    Matthews, M.M., Sumner, D.P.: Hamiltonian results in K 1, 3-free graphs. J. Graph Theory 8, 139–146 (1984)Google Scholar
  16. 16.
    Plummer, M.D.: Graph factors and factorization: 1985–2003: A survey. Discrete Math. 307, 791–821 (2007)Google Scholar
  17. 17.
    Ryjáček, Z.: On a closure concept in claw-free graphs. J. Comb. Theory Ser. B 70, 217–224 (1997)Google Scholar
  18. 18.
    Ryjáček, Z., Saito, A., Schelp, R.H.: Closure, 2-factors, and cycle coverings in claw-free graphs. J. Graph Theory 32, 109–117 (1999)Google Scholar
  19. 19.
    Yoshimoto, K.: On the number of components in 2-factors of claw-free graphs. Discrete Math. 307, 2808–2819 (2007)Google Scholar

Copyright information

© Springer 2009

Authors and Affiliations

  • Hajo Broersma
    • 1
  • Daniël Paulusma
    • 1
  • Kiyoshi Yoshimoto
    • 2
  1. 1.Department of Computer ScienceDurham University, Science LaboratoriesDurhamEngland
  2. 2.Department of MathematicsCollege of Science and Technology, Nihon UniversityTokyoJapan

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