Equitable coloring and equitable choosability of graphs with small maximum average degree

Abstract

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). 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⌉$\left\lceil {{{\left| {V(G)} \right|} \over k}} \right\rceil$ vertices. In this paper, we prove that if G is a graph such that mad(G) < 3, then G is equitably k-colorable and equitably k- choosable where k ≥ max{Δ(G), 4}. Moreover, if G is a graph such that 125${{12} \over 5}$ , then G is equitably k-colorable and equitably k-choosable where k ≥ max{Δ (G), 3}.