Equitable coloring of sparse planar graphs

TL;DR

This paper establishes bounds on the equitable chromatic threshold for sparse planar graphs with certain girth conditions, advancing understanding of equitable coloring in graph theory.

Contribution

It provides new upper bounds on the equitable chromatic threshold for planar graphs with minimum degree two and specified girth, extending prior results.

Findings

For girth at least 10, the threshold is at most 4.

For girth at least 14, the threshold is at most 3.

These bounds improve understanding of equitable coloring in sparse planar graphs.

Abstract

A proper vertex coloring of a graph $G$ is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold $\chi_{eq}^*(G)$ of $G$ is the smallest integer $m$ such that $G$ is equitably $n$-colorable for all $n\ge m$. We show that for planar graphs $G$ with minimum degree at least two, $\chi_{eq}^*(G)\le 4$ if the girth of $G$ is at least $10$, and $\chi_{eq}^*(G)\le 3$ if the girth of $G$ is at least $14$.