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Journal Articles European Journal of Combinatorics Year : 2013

Acyclic edge-coloring using entropy compression

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Louis Esperet

Abstract

An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G and every cycle contains at least three colors. We prove that every graph with maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4 colors, improving the previous bound of 9.62 (Delta - 1). Our bound results from the analysis of a very simple randomised procedure using the so-called entropy compression method. We show that the expected running time of the procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices and edges of G. Such a randomised procedure running in expected polynomial time was only known to exist in the case where at least 16 Delta colors were available. Our aim here is to make a pedagogic tutorial on how to use these ideas to analyse a broad range of graph coloring problems. As an application, also show that every graph with maximum degree Delta has a star coloring with 2 sqrt(2) Delta^{3/2} + Delta colors.

Dates and versions

hal-00843770 , version 1 (12-07-2013)

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Louis Esperet, Aline Parreau. Acyclic edge-coloring using entropy compression. European Journal of Combinatorics, 2013, 34 (6), pp.1019-1027. ⟨10.1016/j.ejc.2013.02.007⟩. ⟨hal-00843770⟩
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