Injective Coloring Of Plane Graphs

Unless state otherwise, We just talk about simple finite graphs here. A mapping c from the vertex set of G denote by V(G) to the color set {1,2, … , k} such that any two adjacent vertices own different colors is called a proper k- coloring of G. The least k such that G has a proper k- coloring is the chromatic number χ(G) of G. A vertex coloring φ of G is called Injective coloring if any two vertices have a same adjacent vertex and own different colors. Similarly, the Injective chromatic number of G is denoted by χi(G) and the Injective list chromatic number of G is denoted by χil,(G). Obviously, we have △(G)≤ χi(G)≤ △(G)(△(G)-1)+1, (G≠K2) and χi(G)≥χ(G).Since Hahn.etc introduced Injective coloring theorem, people were fervent for re-search about Injective coloring of graphs. In summary, people studied Injective coloring mainly by consider the girth, the maximum degree, as well as the maximum average degree and so on. Of course, predecessors have put forward many important conjec-tures that is one of the factors why many scholars at home and abroad with enthusiasm to scientific research, including the famous Four-color Theorem.In this paper, we talk about the Injective chromatic number without short cycles. Firstly we introduce some background knowledge and research status of Injective col-oring. As a relatively new research orientation of coloring planar graphs, we mainly study the Injective chromatic number of planar graph with grith at least 5.