DP-4-coloring Of Planar Graphs Without 5-cycles Adjacent To 6-cycles Or Intersecting 5-cycles

Graph theory is a branch of mathematics,a subject that has developed rapidly and has been widely used in recent years.The coloring problem is a very active research topic in graph theory,with profound and rich theoretical results and wide practical applications,its theory and methods It occupies an important position in discrete mathematics.In the paper,we prove that every planar graph without 5-cycles adjacent to 6-cycles is DP-4-colorable,and also show that every planar graph without intersecting 5-cycles is DP-4-colorable.The concrete content is in the following:·In Chapter 1,we introduce the background and significance of the research,including the development of a representative at home and abroad regarding this aspect,some related lemmas and the theorems that need to be proved in the paper.Based on this research background and profound discussion,by using deep-going analysis,it fully shows the main work's necessity and innovation.·In Chapter 2,we give some necessary definition,graph structure.·In Chapter 3,in order to prove the theorem 1.3,this chapter proves a stronger result,see the theorem 3.1,and then uses the theorem 3.1 to prove the theo-rem 1.3.Let(G,C0)be a minimal counterexample to Theorem 3.1 with |V(G)|minimized,where C0 is a precolored k-cycle in G,where k=3,4.If C0 is a separating cycle,then any precoloring of C0 can be extend to int(C0)and ext(C0),respectively.Then we get a DP-4-coloring of G,a contradiction.We study some structural features of the minimum counterexample graph G,and find and prove that the graph G doesn't contain the reducible configurations.Then,according to the found reducible configurations,we define suitable dis-charging rules,and use discharging rules to check the final charge of each vertex v and each face f,and get 0<?x?V?F ch'(x)=?x?V?F ch(x)=0,draw a contradiction.We prove that the minimum counterexample graph G doesn't exist,and thus prove the correctness of theorem 1.3.·In Chapter 4,in order to prove the theorem 1.4,this chapter proves a stronger result,see the theorem 4.1,and then uses the theorem 4.1 to prove the theorem 1.4.Let C be a separating 3-cycle in G.By the minimality of(G,C0),any precoloring of C0 can be extended to G-int(C).After that,C is precolored,then again the coloring of C can be extended to int(C).Thus,we get a DP-4-coloring of G,a contradiction.We study some structural features of the minimum counterexample graph G,and find and prove that the graph G doesn't contain the reducible configurations.Then,according to the found reducible configurations,we define suitable discharging rules,similar to Chapter 3,and use discharging rules to check the final charge,draw a contradiction.Prove that the minimum counterexample graph G doesn't exist,thus proving the theorem 1.4 correctness.·In Chapter 5,we summarize the main results in this paper and give some prospects for further research in the future.