An algorithmic answer to the ore-type version of Dirac’s question on disjoint cycles

Henry Kierstead, A. V. Kostochka, T. Molla, D. Yager

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Corrádi and Hajnal in 1963 proved the following theorem on the NP-complete problem on the existence of k disjoint cycles in an n-vertex graph G: For all k ≥ 1 and n ≥ 3k, every (simple) n-vertex graph G with minimum degree δ(G) ≥ 2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k − 1)-connected multigraphs do not contain k disjoint cycles? Recently, Kierstead, Kostochka, and Yeager resolved this question. In this paper, we sharpen this result by presenting a description that can be checked in polynomial time of all multigraphs G with no k disjoint cycles for which the underlying simple graph G̲ satisfies the following Ore-type condition: dG̲(v)+dG̲(u)≥4k−3 for all nonadjacent u, v ∈ V (G).

Original languageEnglish (US)
Title of host publicationSpringer Optimization and Its Applications
PublisherSpringer International Publishing
Pages149-168
Number of pages20
DOIs
StatePublished - Jan 1 2018

Publication series

NameSpringer Optimization and Its Applications
Volume139
ISSN (Print)1931-6828
ISSN (Electronic)1931-6836

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ASJC Scopus subject areas

  • Control and Optimization

Cite this

Kierstead, H., Kostochka, A. V., Molla, T., & Yager, D. (2018). An algorithmic answer to the ore-type version of Dirac’s question on disjoint cycles. In Springer Optimization and Its Applications (pp. 149-168). (Springer Optimization and Its Applications; Vol. 139). Springer International Publishing. https://doi.org/10.1007/978-3-319-94830-0_8