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 V 1 , V 2 , … , V k such that | | V i | − | V j | | ≤ 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.