On r-equitable chromatic threshold of Kronecker products of complete graphs

TL;DR

This paper determines the exact $r$-equitable chromatic threshold for Kronecker products of complete graphs, revealing conditions under which these products share the same colorability as certain complete multipartite graphs.

Contribution

It provides a complete characterization of the $r$-equitable chromatic threshold for $K_m \times K_n$, a problem previously unresolved for general parameters.

Findings

Exact values of $\chi_{r=}^*(K_m \times K_n)$ are derived.

For $r \ge 2$, certain bounds on $n$ ensure $K_m \times K_n$ and $K_{m(n)}$ have identical $r$-equitable colorability.

The results unify and extend understanding of equitable colorings in Kronecker products.

Abstract

A graph $G$ is $r$-equitably $k$-colorable if its vertex set can be partitioned into $k$ independent sets, any two of which differ in size by at most $r$. The $r$-equitable chromatic threshold of a graph $G$, denoted by $\chi_{r=}^*(G)$, is the minimum $k$ such that $G$ is $r$-equitably $k'$-colorable for all $k'\ge k$. Let $G\times H$ denote the Kronecker product of graphs $G$ and $H$. In this paper, we completely determine the exact value of $\chi_{r=}^*(K_m\times K_n)$ for general $m,n$ and $r$. As a consequence, we show that for $r\ge 2$, if $n\ge \frac{1}{r-1}(m+r)(m+2r-1)$ then $K_m\times K_n$ and its spanning supergraph $K_{m(n)}$ have the same $r$-equitable colorability, and in particular $\chi_{r=}^*(K_m\times K_n)=\chi_{r=}^*(K_{m(n)})$, where $K_{m(n)}$ is the complete $m$-partite graph with $n$ vertices in each part.