Planar Graphs Without 5-,6-cycles And Triangle 7-cycles And With Distance Of Triangles At Least 2 Are DP-3-colorable

DP-coloring was introduced by Dvorak and Postle.Let G be a simple graph with n vertices,for each v E V(G),a list assignment L give v a list L(v)of available colors,let Lv={v} x L(v).Let Muv be a match between Lv and Lv(impossible empty),and let ML={Muv:uv E E(G)} be a match assignment of G.Let GL be a graph that satisfies V(GL)={(v,c):v ?V(G),c E L(v)} and E(GL)=?v?V(G)E(GL[Lv])? ML,in which GL[Lv]is a complete graph.If GL contains an independent set of size n,then G have a ML-coloring.If for each L satisfies |L(v)|=k(v ? V(G))and each match assignment ML over L,G have a ML-coloring,then G is DP-k-colorable.Dvorak and Postle used DP-coloring to solve Borodin's conjecture that every planar graph without cycles of length 4 to 8 is 3-choosable.A a generalization of list-coloring,since DP-coloring was introduced,many scholars have extended some conclusions of list-coloring to DP-coloring.In this paper,we will also do such research.Panpan Cheng,Yingqian Wang and Min Chen proved that planar graphs without 5-,6-cycles and triangle 7-cycles and d?? 2 are 3-choosable.We generalize this conclusion to DP-3-coloring,stating that planar graphs without 5-,6-cycles and triangle 7-cycles and d??2 are DP-3-coloring.Because each pre-coloring on some subgraphs can be extended to the whole graph,in this paper,we actually prove a stronger statement:let G be a planar graph that contain no 5-,6-cycles and triangle 7-cycles and d??2,let C0 be a 3-,4-,7-,8-,9-,10-or 11-cycle,then each DP-3-coloring of C0 can be extended to G.The main theorem of this paper is proved by counterproof method,and the idea of vertex minimal counterexample is used.The proof process is mainly divided into two parts:first,analyzing the structure of the vertex minimal counterexample graph,identification of vertices is used in the process of proving the reducible structure.Next,using discharging method,we give the initial charge of vertices and faces and appropriate discharging rules,according to this rules,we analyze the final charge and obtain the sum of initial charge is different from the sum of final charge,which is a contradiction.The proof is completed.