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 \(k \geq \max\{\Delta(G), 6\}\).