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Graphs and Combinatorics

, Volume 12, Issue 4, pp 327–331 | Cite as

Perfect matchings in regular bipartite graphs

  • P. Katerinis
  • N. Tsikopoulos
Article

Abstract

P. Hall, [2], gave necessary and sufficient conditions for a bipartite graph to have a perfect matching. Koning, [3], proved that such a graph can be decomposed intok edge-disjoint perfect matchings if and only if it isk-regular. It immediately follows that in ak-regular bipartite graphG, the deletion of any setS of at mostk − 1 edges leaves intact one of those perfect matchings. However, it is not known what happens if we delete more thank − 1 edges. In this paper we give sufficient conditions so that by deleting a setS ofk + r edgesr ≥ 0, stillG − S has a perfect matching. Furthermore we prove that our result, in some sense, is best possible.

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References

  1. 1.
    Bondy, J.A., Murty, U.S.R., Graph theory with applications, North Holland (1976)Google Scholar
  2. 2.
    Hall, P.: On Representatives of Subsets. J. London Math. Soc.10, 26–30 (1935)Google Scholar
  3. 3.
    Koning, D.: Uber Graphen und ihre anwendung auf determinanten theorie und mengenlehre. Mathematich Annalen77, 453–465 (1976)CrossRefGoogle Scholar
  4. 4.
    Referee II of initial manuscript, Personal communicationGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • P. Katerinis
    • 1
  • N. Tsikopoulos
    • 1
  1. 1.Department of InformaticsAthens University of EconomicsAthensGreece

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