Equitable Coloring and Equitable Choosability of Planar Graphs without $6$- and $7$-Cycles

Abstract

For any given $k$-uniform list assignment $L$, a graph $G$ is equitably $k$-choosable if and only if $G$ is $\ell$-colorable and each color appears on at most $\lceil \frac{|V(G)|}{k}\rceil$ vertices. A graph $G$ is equitably $\ell$-colorable if $G$ has a proper vertex coloring with $k$ colors such that the size of the color classes differ by at most $1$. In this paper, we prove that every planar graph $G$ without $6$- and $7$-cycles is equitably $k$-colorable and equitably $k$-choosable where $k\ge max\{\mathrm{\Delta }(G),6\}$.