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

TL;DR

This paper determines the exact equitable chromatic number and threshold for the Kronecker product of complete multipartite graphs and complete graphs when the sum of the part sizes is at most n.

Contribution

It provides exact formulas for the equitable chromatic number and threshold of specific graph products, advancing understanding in graph coloring.

Findings

Exact values of $ ext{chi}_=(K_{m_1,..., m_r} imes K_n)$ and $ ext{chi}_=^*(K_{m_1,..., m_r} imes K_n)$ are derived.

Results apply when the sum of the part sizes is less than or equal to n.

Enhances knowledge of equitable coloring in graph products.

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 $\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The equitable chromatic threshold of a graph $G$, denoted by $\chi_=^*(G)$, is the minimum $t$ such that $G$ is equitably $k$-colorable for $k \ge t$. In this paper, we give the exact values of $\chi_=(K_{m_1,..., m_r} \times K_n)$ and $\chi_=^*(K_{m_1,..., m_r} \times K_n)$ for $\sum_{i = 1}^r m_i \leq n$.