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

Automated summary

In this paper, the authors extend previous studies on dynamic programming coloring and list coloring by considering different types of cycles and their adjacency relations. They present a theorem for every planar graph and provide a generalization of previous results.