Equitable Coloring and Equitable Choosability of Planar Graphs without chordal 4-and 6-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 [vertical bar V (G)vertical bar/k] vertices. A graph is equitably k-colorable if the vertex set V (G) can be partitioned into k independent subsets V-1, V-2,center dot center dot center dot,V-k such that vertical bar vertical bar V-i vertical bar - vertical bar V-j vertical bar vertical bar <= 1 for 1 <= i; j <= k. In this paper, we prove that if G is a planar graph without chordal 4- and 6-cycles, then G is equitably k-colorable and equitably k-choosable where k >= max{Delta(G); 7}.