e-injective coloring: injective and 2-distance colorings conjectures

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 a path P4 in a graph G, then they are assigned with different labels. With this new definition, we want to take a review at injective coloring of a graph from the new point of view. For this purpose, we will have a comparison between e-injective coloring with usual coloring, injective coloring, and 2-distance coloring. As well, we review the conjectures raised so far in the literature of injective coloring and 2-distance coloring, from the new approach, e-injective coloring. Finally, we precisely investigate the e-injective coloring of trees, join of two graphs, a family of standard graphs, grid graphs, cylinder graphs and tori graphs.