A fast algorithm for equitable coloring

Abstract

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. The celebrated Hajnal-Szemerédi Theorem states: For every positive integer r, every graph with maximum degree at most r has an equitable coloring with r + 1 colors. We show that this coloring can be obtained in O(rn 2 ) time, where n is the number of vertices.

Extremal graph packing problems: Ore-type versus Dirac-type

“…A shorter proof appeared in [38]. Then Kierstead, Kostochka, Mydlarz, and Szemerédi [45] devised an algorithm that in time O(rn 2 ) finds an equitable (r+1)-coloring for any n-vertex graph with maximum degree at most r. It is based on a modification of the proof of Theorem 5.1 in [38]. Here we present a yet shorter and simpler proof of Theorem 5.1.…”

Equitable Coloring of Graphs. Recent Theoretical Results and New Practical Algorithms

“…This conjecture was proved in 1970 by Hajnal and Szemerédi [17] with a long and complicated proof. Kierstead et al [23] found a polynomial-time algorithm of complexity O(∆|V (G)| 2 ) for such a coloring. Kierstead and Kostochka [21] gave a short proof of this theorem, and presented another polynomial-time algorithm for such a coloring.…”

Backtracking based iterated tabu search for equitable coloring

“…Méndez-Díaz et al investigated a polyhedral approach [23], a DSatur-based exact algorithm [25], and a tabu search heuristic [24]. Kierstead et al proposed a fast algorithm (O(n 2 )) to find an equitable k-coloring where k = ∆(G) + 1 [19].…”