The equitable chromatic number of a graph is the smallest integer $n$ for which the graph's vertex set can be partitioned into $n$ independent sets, each pair of which differs in size by at most 1. We develop a formula and a linear-time algorithm which compute the equitable chromatic number of an arbitrary complete multipartite graph. These results yield tractable solutions of certain scheduling problems.
"Equitable Chromatic Number of Complete Multipartite Graphs." Missouri J. Math. Sci. 15 (2) 75 - 81, Spring 2003. https://doi.org/10.35834/2003/1502075