Minimum cost edge-colorings of trees can be reduced to matchings

Takehiro Ito, Naoki Sakamoto, Xiao Zhou, Takao Nishizeki

Research output: Contribution to journalArticlepeer-review


Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω( f ) of an edge-coloring f of G is the sum of costs ω( f (e)) of colors f (e) assigned to all edges e in G. An edge-coloring f of G is optimal if ω( f ) is minimum among all edge-colorings of G. In this paper, we show that the problem of finding an optimal edge-coloring of a tree T can be simply reduced in polynomial time to the minimum weight perfect matching problem for a new bipartite graph constructed from T. The reduction immediately yields an efficient simple algorithm to find an optimal edge-coloring of T in time O(n1.5Δ log(nNω)), where n is the number of vertices in T, Δ is the maximum degree of T, and Nω is the maximum absolute cost |ω(c)| of colors c in C. We then show that our result can be extended for multitrees.

Original languageEnglish
Pages (from-to)190-195
Number of pages6
JournalIEICE Transactions on Information and Systems
Issue number2
Publication statusPublished - 2011 Feb


  • Algorithm
  • Cost edge-coloring
  • Multitree
  • Perfect matching
  • Tree


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