Equitable Coloring and Equitable Choosability of Planar Graphs Without 5- and 7-Cycles

A graph G is equitably k-choosable if for any k-uniform list assignment L, G is L-colorable and each color appears on at most [|V(G)|/k] vertices. A graph G is equitable k-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 if G is a planar graph without 5- and 7-cycles, then G is equitably k-choosable and equitably k-colorable where k≥max{Δ(G),7}.