A step toward Chen-Lih-Wu conjecture

An equitable $k$-coloring of a graph is a proper $k$-coloring where the sizes of any two different color classes differ by at most one. In 1973, Meyer conjectured that every connected graph $G$ has an equitable $k$-coloring for some $k\leq \Delta(G)$, unless $G$ is a complete graph or an odd cycle. Chen, Lih, and Wu strengthened this in 1994 by conjecturing that for $k\geq 3$, the only connected graphs of maximum degree at most $k$ with no equitable $k$-coloring are the complete bipartite graph $K_{k,k}$ for odd $k$ and the complete graph $K_{k+1}$. A more refined conjecture was proposed by Kierstead and Kostochka, relaxing the maximum degree condition to an Ore-type condition. Their conjecture states the following: for $k\geq 3$, if $G$ is an $n$-vertex graph such that $d(x) + d(y)\leq 2k$ for every edge $xy\in E(G)$, and $G$ admits no equitable $k$-coloring, then $G$ contains either $K_{k+1}$ or $K_{m,2k-m}$ for some odd $m$. We prove that for any constant $c>0$ and all sufficiently large $n$, the latter two conjectures hold for every $k\geq cn$. Our proof yields an algorithm with polynomial time that decides whether $G$ has an equitable $k$-coloring, thereby answering a conjecture of Kierstead, Kostochka, Mydlarz, and Szemer\'{e}di when $k \ge cn$.