Equitable coloring of graphs

Abstract

If the vertices of a graph G are colored with k colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then G is said to be equitably k-colorable. The equitable chromatic number x= (G) is the smallest integer k such that G is equitably k-colorable. In the first introduction section, results obtained about the equitable chromatic number before 1990 are surveyed. The research on equitable coloring has attracted enough attention only since the early 1990s. In the subsequent sections, positive evidence for the important equitable A-coloring conjecture is supplied from graph classes such as forests, split graphs, outerplanar graphs, series-parallel graphs, planar graphs, graphs with low degeneracy, graphs with bounded treewidth, Kneser graphs, and interval graphs. Then three kinds of graph products are investigated. A list version of equitable coloring is introduced. The equitable coloring is further examined in the wider context of graph packing. Appropriate conjectures for equitable A-coloring of disconnected graphs are then studied. Variants of the well-known and significant Hajnal and Szemeredi Theorem are discussed. A brief summary of applications of equitable coloring is given. Related notions, such as equitable edge coloring, equitable total coloring, equitable defective coloring, and equitable coloring of uniform hypergraphs, are touched upon. This chapter ends with a short conclusion section. This survey is an updated version of Lih [102].