Equitable coloring of sparse planar graphs

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 $χ_{eq}^*(G)$ 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, $χ_{eq}^*(G)\le 4$ if the girth of $G$ is at least $10$, and $χ_{eq}^*(G)\le 3$ if the girth of $G$ is at least $14$.

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