On Equitable Coloring of d-Degenerate Graphs

Abstract

Abstract. An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most 1. A d-degenerate graph is a graph G in which every subgraph has a vertex with degree at most d. A star Sm with m rays is an example of a 1-degenerate graph with maximum degree m that needs at least 1 + m/2 colors for an equitable coloring. Our main result is that every n-vertex d-degenerate graph G with maximum degree at most n/15 can be equitably k-colored for each k ≥ 16d. The proof of this bound is constructive. We extend the algorithm implied in the proof to an O(d)-factor approximation algorithm for equitable coloring of an arbitrary d-degenerate graph. Among the implications of this result is an O(1)-factor approximation algorithm for equitable coloring of planar graphs with fewest colors. A variation of equitable coloring (equitable partitions) is also discussed.

The Randomized Coloring Procedure with Symmetry-Breaking

“…Note that since a graph with maximum degree ∆ is a ∆-inductive graph, result (E3) can be viewed as a generalization of result (E1), with the small change that the leading constant is 1/e instead of 1 − 1/e. This result extends known results on classical equitable colorings of d-inductive graphs [4,19]. Bollobas and Guy [4] consider the equitable coloring of 1-inductive graphs (i.e., forests) whereas Kostochka et al [19] consider d-inductive graphs for arbitrary d. Specifically, Kostochka et al [19] show that every d-inductive graph has an equitable coloring with at most 16d colors (provided ∆ < n/15).…”

Equitable coloring of random graphs

“…Lih's paper [14] surveys some basic results on equitable colorings and how the bound of ∆ + 1 can be replaced by ∆ for certain classes of graphs. Applications of the Hajnal-Szemerédi theorem and recent results on equitable colorings of graphs can be found in (among others) [1], [2], [9], [11], [12], [19]. Equitable coloring turned out to be useful in establishing bounds on tails of sums of dependent variables [6], [8], [18].…”

“…Let us denote these independent sets by I. By Lemma 4.4, the degeneracy number of the graph induced by the vertices not covered is O(np log −1.5 (np)) (and the maximum degree is at most the maximum degree of the original graph, which is at most max{1.01np, log 2 n} ≪ n log −1.5 (np) by the assumption on p), so by Theorem 2.1 (Kostochka et al [11]), we can color them equitably with as many colors as the number of independent sets in I. In such a way we get independent sets J 1 , J 2 , ..., J |I| ,…”

“…Theorem 2.1 (Kostochka, Nakprasit, Pemmaraju [11]) For every d, n ≥ 1, if a graph G is d-degenerate, has n vertices and satisfies ∆(G) ≤ n/15, then χ * = (G) ≤ 16d. Let us collect a couple of facts, easy inequalities that will be used during the proofs.…”

Packing d-degenerate graphs

“…Note that the graph G 1 is 1-degenerate. The idea of the proof of Theorem 2 is a refinement of that used in [11] for a somewhat similar result on equitable coloring, a partial case of the packing problem.…”