Equitable list colorings of planar graphs without short cycles

Abstract

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 equitably k -colorable if G has a proper k -vertex coloring such that the sizes of any two color classes differ by at most 1. In this paper, we prove that every planar graph G is equitably k -choosable and equitably k -colorable if one of the following conditions holds: (1) G is triangle-free and k ≥ max { Δ ( G ) , 8 } ; (2) G has no 4- and 5-cycles and k ≥ max { Δ ( G ) , 7 } .