An Efficient Algorithm for the Nearly Equitable Edge Coloring Problem

Abstract

An edge coloring of a multigraph is nearly equitable if, among the edges incident to each vertex, the numbers of edges colored with any two colors differ by at most two. It has been proved that the problem of finding a nearly equitable edge coloring can be solved in O(m/k) time, where m and k are the numbers of edges and given colors, respectively. In this paper, we present a recursive algorithm that runs in O (mn log (m/(kn) + 1)) time, where n is the number of vertices. This algorithm improves the bestknown worst-case time complexity. When k = O(1), the time complexity of all known algorithms is O(m), which implies that this time complexity remains to be the best for more than twenty years since 1982 when Hilton and de Werra gave a constructive proof for the existence of a nearly equitable edge coloring for any multigraph. Our result is the first that improves this time complexity when m/n grows to infinity; e.g., m = n for an arbitrary constant θ > 1. Submitted: July 2008 Reviewed: September 2008 Revised: October 2008 Accepted: October 2008 Final: November 2008 Published: December 2008 Article type: Regular paper Communicated by: D. Wagner This research was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan. A preliminary version of this paper appeared in Proceedings of ISAAC [10]. E-mail addresses: sharryx@al.cm.is.nagoya-u.ac.jp (Xuzhen Xie) yagiura@nagoya-u.jp (Mutsunori Yagiura) takao@al.cm.is.nagoya-u.ac.jp (Takao Ono) hirata@is.nagoya-u.ac.jp (Tomio Hirata) zwick@tau.ac.il (Uri Zwick) 384 X. Xie et al. An Efficient Algorithm for Nearly Equitable Edge Coloring