Bicolored Matchings in Some Classes of Graphs

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 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.

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Correspondence to B. Ries.

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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

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Keywords

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