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 x, y. 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 x, y, 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.
Similar content being viewed by others
References
Alon, N., Füredi, Z.: Spanning subgraphs of random graphs. Graphs Comb. 8, 91–94 (1992)
Alon, N., Yuster, R.: \(H\)-factors in dense graphs. J. Comb. Theory Ser. B 66, 269–282 (1996)
Blazewicz, J., Ecker, K., Pesch, E., Schmidt, G., Weglarz, J.: Scheduling Computer and Manufacturing Processes, 2nd edn. Springer, Berlin (2001)
Chen, B.-L., Lih, K.-W., Wu, P.-L.: Equitable coloring and the maximum degree. Eur. J. Comb. 15, 443–447 (1994)
Corrádi, K., Hajnal, A.: On the maximal number of independent circuits in a graph. Acta Math. Acad. Sci. Hung. 14, 423–439 (1963)
Enomoto, H.: On the existence of disjoint cycles in a graph. Combinatorica 18, 487–492 (1998)
Hajnal, A., Szemerédi, E.: Proof of a conjecture of P. Erdős. Combinatorial Theory and its Application, pp. 601–623. North-Holland, London (1970)
Kierstead, H.A., Kostochka, A.V.: An Ore-type theorem on equitable coloring. J. Comb. Theory Ser. B 98, 226–234 (2008)
Kierstead, H.A., Kostochka, A.V.: Ore-type versions of Brooks’ theorem. J. Comb. Theory Ser. B 99, 298–305 (2009)
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)
Kierstead, H.A., Kostochka, A.V.: A refinement of a result of Corrádi and Hajnal. Combinatorica 35, 497–512 (2015)
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)
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
Kierstead, H., Rabern, L.: Personal communication
Kostochka, A.V., Rabern, L., Stiebitz, M.: Graphs with chromatic number close to maximum degree. Discret. Math. 312, 1273–1281 (2012)
Lih, L.-W., Wu, P.-L.: On equitable coloring of bipartite graphs. Discret. Math. 151, 155–160 (1996)
Postle, L.: Personal communication
Rabern, L.: \(\Delta \)-critical graphs with small high vertex cliques. J. Comb. Theory Ser. B 102, 126–130 (2012)
Rödl, V., Ruciński, A.: Perfect matchings in \(\epsilon \)-regular graphs and the blow-up lemma. Combinatorica 19, 437–452 (1999)
Smith, B.F., Bjorstad, P.E., Gropp, W.D.: Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996)
Wang, H.: On the maximum number of disjoint cycles in a graph. Discret. Math. 205, 183–190 (1999)
Yap, H.-P., Zhang, Y.: The equitable \(\Delta \)-colouring conjecture holds for outerplanar graphs. Bull. Inst. Math. Acad. Sin. 5, 143–149 (1997)
Yap, H.-P., Zhang, Y.: Equitable colorings of planar graphs. J. Comb. Math. Comb. Comput. 27, 97–105 (1998)
Acknowledgements
We thank a referee for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Rudolf Halin.
Research of H. A. Kierstead is supported in part by NSA Grant H98230-12-1-0212. Research of A. V. Kostochka is supported in part by NSF Grants DMS-1266016 and DMS-1600592 and by Grant NSh.1939.2014.1 of the President of Russia for Leading Scientific Schools. Research of T. Molla is supported in part by NSF Grant DMS-1500121. Research of E. C. Yeager is supported in part by NSF Grants DMS 08-38434 and DMS-1266016.
Rights and permissions
About this article
Cite this article
Kierstead, H.A., Kostochka, A.V., Molla, T. et al. Sharpening an Ore-type version of the Corrádi–Hajnal theorem. Abh. Math. Semin. Univ. Hambg. 87, 299–335 (2017). https://doi.org/10.1007/s12188-016-0168-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-016-0168-8