Let G = (V (G);E(G)) be a graph, and f be a map from V (G) to a set of labels (colors). The map f is said to be a 2-distance coloring of G if the colors of the vertices of any P3 path are different under f. An injective coloring of a given graph G is a vertex coloring f of G such that no vertex v is adjacent to two vertices u and w with f(u) = f(w). It has been shown that the injective coloring of G, is not necessarily a proper coloring, and vice versa. A map f ∶ V (G) {1; 2; 3; :::; k} is said to be a 2-distance injective k coloring of G if for a vertex v and the vertices u;w; x; y; z; t, there exist a uvw-path of length 2 or a xyvzt path of length 4, then the vertices u;w; x; y; z; t receive distinct colors. In other word, if we observe a path P3 = uvw, or a path P5 = xyvzt in graph, then f(u); f(w); f(x); f(y); f(z) and f(t) are mutually distinct. The 2-distance injective chromatic number of G, denoted by 2i(G), is the least k such that G has a 2-distance injective k-coloring. In this talk we investigate some properties of 2-distance injective coloring of a graph G. The 2- distance injective coloring versus to 2-distance coloring and injective coloring are investigated. Also 2- distance injective coloring are studied in terms of other parameters such as order, degree, independence number and etc.
