a generalization of injective coloring

An injective coloring of a given graph g = (v, e) is a vertex coloring of g such that any two vertices with a common neighbor receive distinct colors. a 2-distance coloring of g is a proper vertex coloring of the graph g such that any pair of vertices at a distance of at most 2 receive different colors. a 2-distance injective coloring of a graph g is a vertex coloring in which any two vertices that share a common neighbor are assigned different colors, and any two vertices that share a common 2-distance neighbor are also assigned different colors. equivalently, for every path p3 = ywz or p5 = xywzt in g, the vertices x, t, and w receive pairwise distinct colors, and y and z are assigned different colors. this new definition provides a generalization of the injective coloring and allows us to revisit the concept from a new perspective. we also investigate the 2-distance injective coloring of several families of graphs.