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چکیده
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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 in which, any two vertices v, u with common edge e (e ̸= uv) receive distinct colors, in other words, any two end vertices of a path P4 of G achieve dierent colors. With this new de nition, we want to take a review at injective coloring of a graph from the new point of view. For this purpose, we review the conjectures raised so far in the literature of injective coloring and 2-distance coloring, from the new approach, e-injective coloring. As well, we prove that, for disjoint graphs G,H, with E(G) ̸= ∅ and E(H) ̸= ∅, χei(G ∪ H) = max{χei(G), χei(H)} and χei(G ∨ H) = |V (G)|+|V (H)|. The e-injective chromatic number of G versus of the maximum degree and packing number of G is investigated, and we denote max{χei(G), χei(H)} ≤ χei(G□H) ≤ χ2(G)χ2(H). Finally, we prove that, for any tree T (T is not a star), χei(T) = χ(T), and we obtain the exact value of e-injective chromatic number of some speci ed graphs.
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