Kierstead-Kostochka Ore-type analogue of the Chen-Lih-Wu Conjecture

Let an equitable $k$-coloring be a proper $k$-coloring whose color classes have sizes differing by at most one, and let $d(x)$ denote the degree of a vertex $x$. Kierstead-Kostochka conjecture. For $k≥3$, if $G$ is a graph satisfying $d(x)+d(y)≤2k$ for every edge $xy$, and $G$ has no equitable $k$-coloring, then $G$ contains either $K_{k+1}$ or $K_{m,2k-m}$ for some odd $m$.

This is the Ore-type analogue of the Chen-Lih-Wu Conjecture. The paper proves it when $k$ is at least a fixed positive proportion of the number of vertices, while the case $k=o(n)$ remains a stated direction for future work.