Equitable and list equitable colorings of planar graphs without 4-cycles

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 ⌈|V(G)|k⌉ vertices. A graph is said to be equitably k-colorable if the vertex set V(G) can be partitioned into k independent subsets V1,V2,…,Vk such that ||Vi|−|Vj||≤1(1≤i,j≤k). In this paper, we prove that every planar graph G without 4-cycles is equitably k-choosable and equitably k-colorable where k≥max{Δ(G),6}, which improves several recent results.