Acyclic edge-coloring using entropy compression

TL;DR

This paper introduces a new bound for acyclic edge-coloring of graphs using entropy compression, reducing the number of colors needed and providing a polynomial-time randomized algorithm with practical implications.

Contribution

It improves the upper bound for acyclic edge-coloring and demonstrates the use of entropy compression in analyzing graph coloring problems.

Findings

Acyclic edge-coloring with at most 4 Delta - 4 colors.

Expected running time of the coloring algorithm is polynomial.

Application to star coloring with specific color bounds.

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.