The (2k-1)-connected multigraphs with at most k-1 disjoint cycles

Abstract

In 1963, Corradi and Hajnal proved that for all k≥1 and n≥3k, every (simple) graph G on n vertices 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, the authors characterized the simple graphs G with minimum degree δ(G)≥2k—1 that do not contain k disjoint cycles. We use this result to answer Dirac's question in full.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    K. Corráadi and A. Hajnal: On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hungar. 14 (1963), 423–439.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    G. Dirac: Some results concerning the structure of graphs, Canad. Math. Bull. 6 (1963), 183–210.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    G. Dirac and P. Erdős: On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hungar. 14 (1963), 79–94.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    H. A. Kierstead, A. V. Kostochka and E. C. Yeager: On the Corradi-Hajnal Theorem and a question of Dirac, submitted.

  5. [5]

    L. Lovasz: On graphs not containing independent circuits, (Hungarian, English summary) Mat. Lapok 16 (1965), 289–299.

    MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Elyse C. Yeager.

Additional information

The first two authors thank Institut Mittag-Leffler (Djursholm, Sweden) for the hospitality and creative environment.

Research of this author is supported in part by NSA grant H98230-12-1-0212.

Research of this author is supported in part by NSF grant DMS-1266016 and by Grant NSh.1939.2014.1 of the President of Russia for Leading Scientific Schools.

Research of this author is supported in part by NSF grants DMS 08-38434 and DMS-1266016.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kierstead, H.A., Kostochka, A.V. & Yeager, E.C. The (2k-1)-connected multigraphs with at most k-1 disjoint cycles. Combinatorica 37, 77–86 (2017). https://doi.org/10.1007/s00493-015-3291-8

Download citation

Mathematics Subject Classification (2000)

  • 05C15
  • 05C35
  • 05C40