DP-3-coloring Of Planar Graphs

Proper coloring refers to the assignment of color sets to each vertex in graph G,so that any two adjacent vertices are dyed with different colors.With the finding and development of the Four Color Theorem,people have defined a variety of improper coloring methods,such as list-coloring.Recently,Dvorak and Postle introduced a new coloring method called DP-coloring,which can be regarded as a generalization of list-coloring.The essence of coloring is to change each vertex into.a clique with the number of list assignment,then assigns an arbitrary matching between lists of colors at adjacent vertices,finally find an independent set among them.Although not all list colorings can be generalized to DP-coloring,such as even circles,DP-coloring is still an effective research method.In fact,there are many conclusions of list-coloring that have been generalized to DP-coloring,among which the more famous ones are:planar graphs without 3-cycles and 4-cycles are DP-3-colorable.People are very interested in the sufficient conditions for DP-3-colorable planar graphs.This article mainly proves the following two theorems:(1)Planar graphs without 4-,6-cycles,excluding adjacent 5-cycles,and the distance between 3-cycles d?? 3 are DP-3-colorable.(2)Planar graphs without 5-,6-cycles,and the distance between 4-cycles d?? 3,the distance between 3-cycles d?? 3 are DP-3-colorable.The research method in this paper is first to be assumed that there is a minimum counterexample satisfied the theorem conditions,study the structural characteristics of the minimum counterexample,then assign weights to its vertices and faces,then formulate discharging rules.The contradiction is obtained through the process of value transfer and immutability.So the minimum counterexample does not exist,the correctness of the theorem is finally proved.