An Ore-type theorem on equitable coloring

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 xy∈E(G), the sum d(x)+d(y) of the degrees of its ends is at most 2r+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.