Strengthening Theorems of Dirac and Erdős on Disjoint Cycles

Henry Kierstead, A. V. Kostochka, A. Mcconvey

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

Let k ≥ 3 be an integer, Hk(G) be the set of vertices of degree at least 2k in a graph G, and Lk(G) be the set of vertices of degree at most 2k−2 in G. In 1963, Dirac and Erdős proved that G contains k (vertex) disjoint cycles whenever |Hk(G)|−|Lk(G)|≥k2+2k−4. The main result of this article is that for k≥2, every graph G with |V(G)|≥3k containing at most t disjoint triangles and with |Hk(G)|−|Lk(G)|≥2k + t contains k disjoint cycles. This yields that if |Hk(G)|−|Lk(G)|≥3k and k≥2, then G contains k disjoint cycles. This generalizes the Corrádi–Hajnal Theorem, which states that every graph G with Hk(G) = V(G) and |Hk(G)|≥3k contains k disjoint cycles.

Original languageEnglish (US)
Pages (from-to)788-802
Number of pages15
JournalJournal of Graph Theory
Volume85
Issue number4
DOIs
StatePublished - Aug 2017

Keywords

  • disjoint cycles
  • disjoint triangles
  • minimum degree
  • planar graphs

ASJC Scopus subject areas

  • Geometry and Topology

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