Journal of Graph Algorithms and Applications
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An Efficient Algorithm for the Nearly Equitable Edge Coloring Problem
Xuzhen Xie, Mutsunori Yagiura, Takao Ono, Tomio Hirata, and Uri Zwick
Vol. 12, no. 4, pp. 383-399, 2008. Regular paper.
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(m2/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 best-known worst-case time complexity. When k=O(1), the time complexity of all known algorithms is O(m2), 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 ϑ for an arbitrary constant ϑ > 1.
Submitted: July 2008.
Reviewed: September 2008.
Revised: October 2008.
Accepted: October 2008.
Final: November 2008.
Published: December 2008.
Communicated by Dorothea Wagner