New bounds for the nearly equitable edge coloring problem

Xuzhen Xie*, Mutsunori Yagiura, Takao Ono, Tomio Hirata, Uri Zwick

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


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 this problem can be solved in O(m 2/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 worstcase 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 graph. Our result is the first that improves this time complexity when m/n grows to infinity; e.g., m = nθ for an arbitrary constant θ > 1. We also propose a very simple randomized algorithm that runs in O (m3/2n1/2/k 1/2) time with probability at least 1 -1/c for any constant c > 1, whose worst-case time complexity is O(m2/k).

Original languageEnglish
Title of host publicationAlgorithms and Computation - 18th International Symposium, ISAAC 2007, Proceedings
PublisherSpringer Verlag
Number of pages12
ISBN (Print)9783540771180
StatePublished - 2007
Event18th International Symposium on Algorithms and Computation, ISAAC 2007 - Sendai, Japan
Duration: 17 Dec 200719 Dec 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4835 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference18th International Symposium on Algorithms and Computation, ISAAC 2007


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