An upper bound for the equitable chromatic number of complete n-partite graphs

A proper vertex coloring of a graph $G$ is equitable if the size of color classes differ by at most one. The equitable chromatic number of $G$ is the smallest integer $m$ such that $G$ is equitable m-colorable. In this paper, we derive an upper bound for the equitable chromatic number of complete n-partite graph $K_{p_{1},p_{2}, ... ,p_{n}}$

Equitable list coloring of planar graphs without 4- and 6-cycles

AbstractA graph G is equitably k-choosable if for any k-uniform list assignment L, there exists an L-colorable of G such that each color appears on at most ⌈|V(G)|k⌉ vertices. Kostochka, Pelsmajer and West introduced this notion and conjectured that G is equitably k-choosable for k>Δ(G). We prove this for planar graphs with Δ(G)≥6 and no 4- or 6-cycles