Equitable Colorings of Planar Graphs without Short Cycles

TL;DR

This paper proves the equitable coloring conjecture for various classes of planar graphs, especially those without short cycles, with improved bounds on maximum degree, extending previous results.

Contribution

It establishes the conjecture for $C_3$-free planar graphs with max degree ≥6 and for planar graphs without $C_4$ with max degree ≥7, and for girth ≥6 with max degree ≥5.

Findings

Proves the conjecture for $C_3$-free planar graphs with max degree ≥6.

Establishes the conjecture for planar graphs without $C_4$ with max degree ≥7.

Confirms the conjecture for planar graphs with girth ≥6 and max degree ≥5.

Abstract

An \emph{equitable coloring} of a graph is a proper vertex coloring such that the sizes of every two color classes differ by at most 1. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree $\Delta \geq 2$ has an equitable coloring with $\Delta$ colors, except when $G$ is a complete graph or an odd cycle or $\Delta$ is odd and $G=K_{\Delta,\Delta}.$ Nakprasit proved the conjecture holds for planar graphs with maximum degree at least 9. Zhu and Bu proved that the conjecture holds for every $C_3$-free planar graph with maximum degree at least 8 and for every planar graph without $C_4$ and $C_5$ with maximum degree at least 7. In this paper, we prove that the conjecture holds for planar graphs in various settings, especially for every $C_3$-free planar graph with maximum degree at least 6 and for every planar graph without $C_4$ with maximum degree at least 7, which improve or generalize results on equitable coloring by Zhu and Bu. Moreover, we prove that the conjecture holds for every planar graph of girth at least 6 with maximum degree at least 5.