List Equitable Coloring of Planar Graphs without $4$- and $6$-Cycles when Δ(G)=5

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 |V(G)|/k vertices. In 2009, Li and Bu obtained that for planar graph G, if Δ(G)≥6 and without $4$- and $6$-cycles, then G is Δ(G) list equitably colorable. In order to further prove the conjecture of list equitable coloring, in this paper, we focus on planar graph with Δ(G)=5, and prove that if G is a planar graph without $4$- and $6$-cycles, then G is Δ(G) list equitably colorable.