Equitable and List Equitable Colorings of Planar Graphs Without 5-Cycles

Abstract

A graph G is k list equitably colorable, if for any given k-uniform list assignment L, G is L-colorable and each color appears on at most \(\lceil \frac{|V(G)|}{k}\rceil \) vertices. Kostochka et al. conjectured that if G is a connected graph with maximum degree at least 3, then G is \(\Delta (G)\) list equitably colorable, unless G is a complete graph or is \(K_{k,k}\) for some odd k. An equitable k-coloring c of G is a mapping c from V(G) to \([k]=\{1,2,\ldots ,k\}\) such that \(c(u)\ne c(v)\) for each \(uv\in E(G)\), and for each \(k_i\), \(k_j \in [k]\), \(||\{v|c(v)=k_i\}|-|\{w|c(w)=k_j\}||\le 1\). Chen et al. conjectured that each connected graph with maximum degree \(\Delta \) that is different from the complete graph \(K_{\Delta +1}\), the complete bipartite graph \(K_{\Delta , \Delta }\) and an odd cycle admits an equitable coloring with \(\Delta \) colors. In this paper, we prove that if G is a planar graph without 5-cycles, then G is k list equitably colorable and equitably k-colorable where \(k\ge \max \{\Delta (G),7\}\).