Equitable chromatic threshold of complete multipartite graphs

TL;DR

This paper introduces a formula and a linear-time algorithm to compute the equitable chromatic threshold of complete multipartite graphs, advancing understanding of equitable colorings in graph theory.

Contribution

It provides the first explicit formula and efficient algorithm for determining the equitable chromatic threshold of complete multipartite graphs.

Findings

Developed a formula for the equitable chromatic threshold.

Designed a linear-time algorithm for computation.

Applied method to arbitrary complete multipartite graphs.

Abstract

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. 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 $k\ge t$. We develop a formula and a linear-time algorithm which compute the equitable chromatic threshold of an arbitrary complete multipartite graph.