A Sufficient Condition For DP-4-colorable Planar Graphs

In this paper,we mainly study the DP coloring problem of planar graphs under certain conditions.In 2017,Postle and others extended the list coloring of planar graphs to DP coloring,which overcomes the limitations of list coloring.This paper is based on Danjun Huang's proof that if a planar graph G has no vertex incident to 3-,4-,5-,6-cycles simultaneously,then G is 4-choosable.The result is extended to DP coloring,we prove that if a planar graph G has no vertex incident to 3-,4-,5-,6-cycles simultaneously,then G is DP-4-colorable,Firstly,we assume that the minimum counterexample of G exists,then we study some structural features of the minimum counterexample,and find some reducible configurations,and define the initial charges for each vertex and each face.By using the handshake lemma and Euler formula,the total sum of the initial charges of vertices and faces is obtained.We then define discharging rules to redistribute charges.Once the discharging procedure is finished,a new charge function will be produced.During the procedure of discharging,the total sum of charges is keep fixed.According to the difference between the final charges and the sum of the initial charges,it is proved that the minimum counterexample of graph G does not exist,which proves the correctness of the conclusion and further develops a new coloring theory.