A Short Proof of the Hajnal-Szemerédi Theorem on Equitable Coloring

An equitable k-coloring of a graph G is a proper k-coloring, for which any two color classes differ in size by at most one. Equitable colorings naturally arise in some scheduling, partitioning, and load balancing problems [1, 15, 16]. Pemmaraju [13] and Janson and Ruciński [6] used equitable colorings to derive deviation bounds for sums of dependent random variables that exhibit limited dependence. In 1964 Erdős [3] conjectured that any graph with maximum degree ∆(G) ≤ r has an equitable (r+ 1)-coloring. This conjecture was proved in 1970 by Hajnal and Szemerédi [5] with a surprisingly long and complicated argument. Recently, Mydlarz and Szemerédi [11] found a polynomial time algorithm for such coloring. In search of an easier proof, Seymour [14] strengthened Erdős’ conjecture by asking whether every graph with minimum degree δ(G) ≥ k k+1 |G| contains the k-th power of a hamiltonian cycle. (If |G| = (r + 1)(s+ 1) and ∆(G) ≤ r then δ(Ḡ) ≥ s s+1 |G|; each (s+ 1)interval of a s-th power of a hamiltonian cycle in Ḡ is an independent set in G.) The case k = 1 is Dirac’s Theorem and the case k = 2 is Pósa’s Conjecture. Fan and Kierstead [4] proved Pósa’s Conjecture with cycle replaced by path. Komlós, Sarkozy and Szemerédi [7] proved Seymour’s conjecture for graphs with sufficiently many (in terms of k) vertices. Neither of these partial results has a simple proof. In fact, [7] uses the Regularity Lemma, the Blow-up Lemma and the Hajnal-Szemerédi Theorem. A different strengthening was suggested recently by Kostochka and Yu [9, 10]. In the spirit of Ore’s theorem on hamiltonian cycles [12], they conjectured that every graph in which d(x) + d(y) ≤ 2r for every edge xy has an equitable (r + 1)-coloring. In this paper we present a short proof of the Hajnal-Szemerédi Theorem and present another polynomial time algorithm that constructs an equitable (r + 1)-coloring of any ∗Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA. E-mail address: [email protected]. Research of this author is supported in part by the NSA grant MDA 904-03-10007 †Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA and Institute of Mathematics, Novosibirsk, 630090, Russia. E-mail address: [email protected]. Research of this author is supported in part by the NSF grant DMS-0400498.