Sufficient Conditions of 6-Cycles Make Planar Graphs DP-4-Colorable

In simple graphs, DP-coloring is a generalization of list coloring and thus many results generalize those coloring. Xu Wu proved that every planar graph without 5-cycles adjacent simultaneously to 3-cycles 4-cycles 4-choosable. Later, Sittitrai Nakprasit showed if has no pairwise 3-, 4-, 5-cycles, then it DP-4-colorable, which the result Wu. this paper, we extend on 5-, 6-cycles by showing 3-cycles, 4-cycles, also previous studies as follows: G DP-4-colorable i-cycles where i∈{3,4,5}.