Every Projective Planar Graph G Without 6-cycles Adjacent To Triangles And M-graph Is DP-4-colorable

DP-coloring was introduced by Dvorak and Postle(2017)as a generalization of list coloring.In this paper,we study DP-coloring of simple finite graphs on the projective plane.DP-coloring is different from proper coloring,which is a very novel direction in graph theory coloring and is favored by many graph theory scholars.As DP-coloring of graph is a generation of list coloring of graph,we define the list assignment L of graph G before the study started,and then define the subgraph H and(L,H)cover.Finally,there is the definition that graph G is(L,H)-coloring.At this time,if f is a mapping from vertices to non-negative integers in G,f:V(G)? Z+,and all vertices v have |L(v)|? f(v),we call the graph G is DP-f-colorable.If we then let |L(v)|=k,and each choice of any cover(L,H)for the graph G can make graph G(L,H)-colorable,the minimum value of k is called the DP-chromatic number of graph G.This is a definition of DP-k-coloring given by Bernshteyn et al,according to the concept of corresponding coloring introduced by Dvorak and Postle.It has been proved that every planar graph G without 6-cycles adjacent to triangles is DP-4-colorable by Liu,Li,et al in 2019.This paper mainly generalizes this conclusion to the projective plane after strengthening the original conditions.For example,it is obvious that a complete graph K5 without 6-cycles adjacent to triangles,which can be embedded on the projective plane.However its chromatic number is 5,and it must not be DP-4-colorable.This paper mainly proves the following results,thereinto M=[v₁v₂v₃v₄],d(v_i)=4(i=1,2,3,4),and the derived subgraph is isomorphic to the complete graph K₄:Theorem 1.Every projective planar graph G without 6-cycles adjacent to triangles and M-graph is DP-4-colorable.