Skip to main content

Bicolored Matchings in Some Classes of Graphs

Graphs and Combinatorics volume 23, pages 47–60 (2007)Cite this article

Abstract

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings M k exist with |M k ∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k.

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

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.: Network flows. Prentice-Hall, 1993

  2. Asratian, A.S., Denley, T.M.J., Häggkvist, R.: Bipartite graphs and their applications. Cambridge University Press, Cambridge, 1998

  3. Berge, C.: Graphes. Gauthier-Villars, Paris, 1983

  4. Chandrasekaran, R., Kaboadi, S.N., Murty, K.G.: Some NP-complete problems in linear programming. Operations Research Letters, 1, 101–104 (1982)

    Google Scholar 

  5. de Werra, D.: On line-perfect graphs. Mathematical programming, 15, 236–238 (1978)

    Google Scholar 

  6. Feige, U., Okek, E., Wieder, U.: Approximating maximum edge coloring in multigraphs. Technical Report, Weizmann Institute, 2003

  7. Gabor, H.N., Tarjan, R.E.: Efficient algorithms for a family of matroid intersection problems. Journal of Algorithms, 5, 80–131 (1984)

    Google Scholar 

  8. Gross, J.L., Yell, J.: Handbook of graph theory. CRC Press, London, 2004

  9. Hartvigsen D.: Extensions of matching theory (Ph.D. thesis). Carnegie–Mellon University, 1984

  10. Holyer, I.: The NP-completeness of edge-coloring. SIAM Journal on Computing, 10, 718–720 (1981)

    Google Scholar 

  11. Karzanov A.V.: Maximum matching of given weight in complete and complete bipartite graphs. Kibernetika, 1:7–11, 1987. English translation in CYBNAW 23, 8–13

    Google Scholar 

  12. Lovasz, L., Plummer, M.D.: Matching Theory. Annals of Discrete Mathematics, 29, 1986

  13. Garey, M.R., Johnson, D.S.: Computers and Intractability, a Guide to the Theory of NP-Completeness. Freeman, New York, 1979

  14. Trotter, L.E.: Line-perfect graphs. Mathematical Programming, 12, 255–259, (1977)

    Google Scholar 

  15. Yi, T., Murty, K.G., Spera, C.: Matchings in colored bipartite networks. Discrete Applied Mathematics, 121, 261–277 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CEDRIC, CNAM, Paris

    M. C. Costa & C. Picouleau

  2. IMA - EPFL, Lausanne

    D. de Werra & B. Ries

Authors
  1. M. C. Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. de Werra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Picouleau
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. B. Ries
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Ries.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Costa, M., Werra, D., Picouleau, C. et al. Bicolored Matchings in Some Classes of Graphs. Graphs and Combinatorics 23, 47–60 (2007). https://doi.org/10.1007/s00373-006-0686-8

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-006-0686-8

Keywords

  • Matchings
  • Alternating cycles
  • Bicolored graphs
  • Cacti
  • bipartite graphs
  • Line-perfect graphs
  • trees