Asymptotically optimal frugal colouring
Journal of Combinatorial Theory Series B
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A basic randomized coloring procedure has been used inprobabilistic proofs to obtain remarkably strong results on graphcoloring. These results include the asymptotic version of the ListColoring Conjecture due to Kahn, the extensions of Brooks' Theoremto sparse graphs due to Kim and Johansson, and Luby's fast paralleland distributed algorithms for graph coloring. The most challengingaspect of a typical probabilistic proof is showing adequateconcentration bounds for key random variables. In this paper, wepresent a simple symmetry-breaking augmentation to the randomizedcoloring procedure that works well in conjunction with Azuma'sMartingale Inequality to easily yield the requisiteconcentration bounds. We use this approach to obtain a number ofresults in two areas: frugal coloring and weightedequitable coloring. A β-frugal coloring of agraph G is a proper vertex-coloring of G in whichno color appears more than β times in anyneighborhood. Let G = (V, E) be avertex-weighted graph with weight function w: V→[0, 1] and let W = Σv εV w(v). A weighted equitablecoloring of G is a proper k-coloring suchthat the total weight of every color class is "large", i.e., "notmuch smaller" than W/k; this notion is useful inobtaining tail bounds for sums of dependent random variables.