Graphs and Combinatorics

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

Bicolored Matchings in Some Classes of Graphs

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≤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 Mk exist with |MkR|≤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.

Keywords

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

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Copyright information

© Springer-Verlag Tokyo 2007

Authors and Affiliations

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

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