Graphs and Combinatorics

, Volume 23, Issue 1, pp 47–60 | Cite as

Bicolored Matchings in Some Classes of Graphs

  • M. C. Costa
  • D. de Werra
  • C. Picouleau
  • B. RiesEmail author


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≤kn. 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 kp. 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.


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.: Network flows. Prentice-Hall, 1993Google Scholar
  2. 2.
    Asratian, A.S., Denley, T.M.J., Häggkvist, R.: Bipartite graphs and their applications. Cambridge University Press, Cambridge, 1998Google Scholar
  3. 3.
    Berge, C.: Graphes. Gauthier-Villars, Paris, 1983Google Scholar
  4. 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. 5.
    de Werra, D.: On line-perfect graphs. Mathematical programming, 15, 236–238 (1978)Google Scholar
  6. 6.
    Feige, U., Okek, E., Wieder, U.: Approximating maximum edge coloring in multigraphs. Technical Report, Weizmann Institute, 2003Google Scholar
  7. 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. 8.
    Gross, J.L., Yell, J.: Handbook of graph theory. CRC Press, London, 2004Google Scholar
  9. 9.
    Hartvigsen D.: Extensions of matching theory (Ph.D. thesis). Carnegie–Mellon University, 1984Google Scholar
  10. 10.
    Holyer, I.: The NP-completeness of edge-coloring. SIAM Journal on Computing, 10, 718–720 (1981)Google Scholar
  11. 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, 8–13Google Scholar
  12. 12.
    Lovasz, L., Plummer, M.D.: Matching Theory. Annals of Discrete Mathematics, 29, 1986Google Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, a Guide to the Theory of NP-Completeness. Freeman, New York, 1979Google Scholar
  14. 14.
    Trotter, L.E.: Line-perfect graphs. Mathematical Programming, 12, 255–259, (1977)Google Scholar
  15. 15.
    Yi, T., Murty, K.G., Spera, C.: Matchings in colored bipartite networks. Discrete Applied Mathematics, 121, 261–277 (2002)Google Scholar

Copyright information

© Springer-Verlag Tokyo 2007

Authors and Affiliations

  • M. C. Costa
    • 1
  • D. de Werra
    • 2
  • C. Picouleau
    • 1
  • B. Ries
    • 2
    Email author
  1. 1.CEDRIC, CNAMParis
  2. 2.IMA - EPFLLausanne

Personalised recommendations