Equitable Coloring and Equitable Choosability of Planar Graphs without chordal 4- and 6-Cycles

TL;DR

This paper proves that certain planar graphs without specific chordal cycles are both equitably k-colorable and k-choosable for k at least the maximum degree or 7, extending understanding of coloring properties.

Contribution

It establishes new bounds for equitable coloring and choosability of planar graphs lacking chordal 4- and 6-cycles, linking structural restrictions to coloring properties.

Findings

Planar graphs without chordal 4- and 6-cycles are equitably k-colorable for k ≥ max{Δ(G), 7}.

Such graphs are also equitably k-choosable under the same conditions.

The results extend previous coloring theories to a broader class of planar graphs.

Abstract

A graph $G$ is equitably $k$-choosable if, for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices. A graph is equitably $k$-colorable if the vertex set $V(G)$ can be partitioned into $k$ independent subsets $V_1$, $V_2$, $\cdots$, $V_k$ such that $||V_i|-|V_j||\leq 1$ for $1\leq i, j\leq k$. In this paper, we prove that if $G$ is a planar graph without chordal $4$- and $6$-cycles, then $G$ is equitably $k$-colorable and equitably $k$-choosable where $k\geq\max\{\Delta(G), 7\}$.