E-injective coloring of graphs

An injective coloring of a given graph G = (V, E) is a vertex coloring of G such that any two vertices with common neighbor receive distinct colors. An e-injective coloring of a graph G is a vertex coloring of G such that any two vertices with common edge neighbor receive distinct colors; in the other words, if u and v are the end of P4 = uxyv in a graph G, then u and v are assigned with different labels. We initiate the study of the concepts of the e-injective coloring for any graph. It is indicated, for any tree T , (other than star), the chromatic number of T and the e-injective chromatic number of T are same. We investigate an e-injective chromatic number of G versus of maximum degree of G and packing number of G. The e-injective chromatic number of union, join and Cartesian product of two graphs are studied. We determine the exact value for e-injective chromatic number of custom graphs. Finally we study the e-injective coloring of the Cartesian product of path Pn and cycle Cn with some specific graphs.