On the Corrádi–Hajnal theorem and a question of Dirac

H. A. Kierstead, A. V. Kostochka, E. C. Yeager

Research output: Contribution to journalArticlepeer-review


In 1963, Corrádi and Hajnal proved that for all k≥1 and n≥3k, every graph G on n vertices with minimum degree δ(G)≥2k contains k disjoint cycles. The bound δ(G)≥2k is sharp. Here we characterize those graphs with δ(G)≥2k−1 that contain k disjoint cycles. This answers the simple-graph case of Dirac's 1963 question on the characterization of (2k−1)-connected graphs with no k disjoint cycles. Enomoto and Wang refined the Corrádi–Hajnal Theorem, proving the following Ore-type version: For all k≥1 and n≥3k, every graph G on n vertices contains k disjoint cycles, provided that d(x)+d(y)≥4k−1 for all distinct nonadjacent vertices x,y. We refine this further for k≥3 and n≥3k+1: If G is a graph on n vertices such that d(x)+d(y)≥4k−3 for all distinct nonadjacent vertices x,y, then G has k vertex-disjoint cycles if and only if the independence number α(G)≤n−2k and G is not one of two small exceptions in the case k=3. We also show how the case k=2 follows from Lovász’ characterization of multigraphs with no two disjoint cycles.

Original languageEnglish (US)
Pages (from-to)121-148
Number of pages28
JournalJournal of Combinatorial Theory. Series B
StatePublished - Jan 1 2017


  • Disjoint cycles
  • Equitable coloring
  • Graph packing
  • Minimum degree
  • Ore-degree

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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