Equitable coloring of Kronecker products of complete multipartite graphs and complete graphs

Abstract

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic number of a graph $G$, denoted by $χ_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The equitable chromatic threshold of a graph $G$, denoted by $χ_=^*(G)$, is the minimum $t$ such that $G$ is equitably $k$-colorable for $k \ge t$. In this paper, we give the exact values of $χ_=(K_{m_1,..., m_r} \times K_n)$ and $χ_=^*(K_{m_1,..., m_r} \times K_n)$ for $\sum_{i = 1}^r m_i \leq n$.

11 pages. arXiv admin note: substantial text overlap with arXiv:1208.0918, arXiv:1207.3578