### Abstract

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(n^{1.5}Δ 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 language | English |
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Pages (from-to) | 190-195 |

Number of pages | 6 |

Journal | IEICE Transactions on Information and Systems |

Volume | E94-D |

Issue number | 2 |

DOIs | |

Publication status | Published - 2011 Feb |

### Keywords

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

### ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Vision and Pattern Recognition
- Electrical and Electronic Engineering
- Artificial Intelligence

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## Cite this

*IEICE Transactions on Information and Systems*,

*E94-D*(2), 190-195. https://doi.org/10.1587/transinf.E94.D.190