An Ore-type theorem on equitable coloring

Abstract

A proper vertex coloring of a graph is equitable if the sizes of its color classes differ by at most one. In this paper, we prove that if G is a graph such that for each edge x y ∈ E ( G ) , the sum d ( x ) + d ( y ) of the degrees of its ends is at most 2 r + 1 , then G has an equitable coloring with r + 1 colors. This extends the Hajnal–Szemerédi Theorem on graphs with maximum degree r and a recent conjecture by Kostochka and Yu. We also pose an Ore-type version of the Chen–Lih–Wu Conjecture and prove a very partial case of it.