Sharpening an Ore-type version of the Corrádi–Hajnal theorem

  • H. A. Kierstead
  • A. V. Kostochka
  • T. Molla
  • E. C. Yeager
Article
  • 36 Downloads

Abstract

Corrádi and Hajnal (Acta Math Acad Sci Hung 14:423–439, 1963) proved that for all \(k\ge 1\) and \(n\ge 3k\), every (simple) graph G on n vertices with minimum degree \(\delta (G)\ge 2k\) contains k disjoint cycles. The degree bound is sharp. Enomoto and Wang proved the following Ore-type refinement of the Corrádi–Hajnal theorem: For all \(k\ge 1\) and \(n\ge 3k\), every graph G on n vertices contains k disjoint cycles, provided that \(d(x)+d(y)\ge 4k-1\) for all distinct nonadjacent vertices xy. Very recently, it was refined for \(k\ge 3\) and \(n\ge 3k+1\): If G is a graph on n vertices such that \(d(x)+d(y)\ge 4k-3\) for all distinct nonadjacent vertices xy, then G has k vertex-disjoint cycles if and only if the independence number \(\alpha (G)\le n-2k\) and G is not one of two small exceptions in the case \(k=3\). But the most difficult case, \(n=3k\), was not handled. In this case, there are more exceptional graphs, the statement is more sophisticated, and some of the proofs do not work. In this paper we resolve this difficult case and obtain the full picture of extremal graphs for the Ore-type version of the Corrádi–Hajnal theorem. Since any k disjoint cycles in a 3k-vertex graph G must be 3-cycles, the existence of such k cycles is equivalent to the existence of an equitable k-coloring of the complement of G. Our proof uses the language of equitable colorings, and our result can be also considered as an Ore-type version of a partial case of the Chen–Lih–Wu Conjecture on equitable colorings.

Keywords

Disjoint cycles Equitable coloring Minimum degree 

Mathematics Subject Classification

05C15 05C35 05C40 

References

  1. 1.
    Alon, N., Füredi, Z.: Spanning subgraphs of random graphs. Graphs Comb. 8, 91–94 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alon, N., Yuster, R.: \(H\)-factors in dense graphs. J. Comb. Theory Ser. B 66, 269–282 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blazewicz, J., Ecker, K., Pesch, E., Schmidt, G., Weglarz, J.: Scheduling Computer and Manufacturing Processes, 2nd edn. Springer, Berlin (2001)CrossRefMATHGoogle Scholar
  4. 4.
    Chen, B.-L., Lih, K.-W., Wu, P.-L.: Equitable coloring and the maximum degree. Eur. J. Comb. 15, 443–447 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Corrádi, K., Hajnal, A.: On the maximal number of independent circuits in a graph. Acta Math. Acad. Sci. Hung. 14, 423–439 (1963)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Enomoto, H.: On the existence of disjoint cycles in a graph. Combinatorica 18, 487–492 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hajnal, A., Szemerédi, E.: Proof of a conjecture of P. Erdős. Combinatorial Theory and its Application, pp. 601–623. North-Holland, London (1970)Google Scholar
  8. 8.
    Kierstead, H.A., Kostochka, A.V.: An Ore-type theorem on equitable coloring. J. Comb. Theory Ser. B 98, 226–234 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kierstead, H.A., Kostochka, A.V.: Ore-type versions of Brooks’ theorem. J. Comb. Theory Ser. B 99, 298–305 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kierstead, H.A., Kostochka, A.V.: Every \(4\)-colorable graph with maximum degree \(4\) has an equitable \(4\)-coloring. J. Graph Theory 71, 31–48 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kierstead, H.A., Kostochka, A.V.: A refinement of a result of Corrádi and Hajnal. Combinatorica 35, 497–512 (2015)Google Scholar
  12. 12.
    Kierstead, H.A., Kostochka, A.V., Yeager, E.C.: On the Corrádi–Hajnal Theorem and a question of Dirac J. Comb. Theory 122, 121–148 (2017)Google Scholar
  13. 13.
    Kierstead, H.A., Kostochka, A.V., Yeager, E.C.: The \((2k-1)\)-connected multigraphs with at most k-1 disjoint cycles. Combinatorica (2015). doi:10.1007/s00493-015-3291-8
  14. 14.
    Kierstead, H., Rabern, L.: Personal communicationGoogle Scholar
  15. 15.
    Kostochka, A.V., Rabern, L., Stiebitz, M.: Graphs with chromatic number close to maximum degree. Discret. Math. 312, 1273–1281 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lih, L.-W., Wu, P.-L.: On equitable coloring of bipartite graphs. Discret. Math. 151, 155–160 (1996)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Postle, L.: Personal communicationGoogle Scholar
  18. 18.
    Rabern, L.: \(\Delta \)-critical graphs with small high vertex cliques. J. Comb. Theory Ser. B 102, 126–130 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Rödl, V., Ruciński, A.: Perfect matchings in \(\epsilon \)-regular graphs and the blow-up lemma. Combinatorica 19, 437–452 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Smith, B.F., Bjorstad, P.E., Gropp, W.D.: Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996)Google Scholar
  21. 21.
    Wang, H.: On the maximum number of disjoint cycles in a graph. Discret. Math. 205, 183–190 (1999)CrossRefMATHGoogle Scholar
  22. 22.
    Yap, H.-P., Zhang, Y.: The equitable \(\Delta \)-colouring conjecture holds for outerplanar graphs. Bull. Inst. Math. Acad. Sin. 5, 143–149 (1997)MathSciNetMATHGoogle Scholar
  23. 23.
    Yap, H.-P., Zhang, Y.: Equitable colorings of planar graphs. J. Comb. Math. Comb. Comput. 27, 97–105 (1998)MathSciNetMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • H. A. Kierstead
    • 1
  • A. V. Kostochka
    • 2
    • 3
  • T. Molla
    • 2
  • E. C. Yeager
    • 2
  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA
  3. 3.Institute of MathematicsNovosibirskRussia

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