Gap vertex-distinguishing edge colorings of graphs

In this paper, we study a new coloring parameter of graphs called the gap vertexdistinguishing edge coloring. It consists in an edge-coloring of a graph G which induces a vertex distinguishing labeling of G such that the label of each vertex is given by the difference between the highest and the lowest colors of its adjacent edges. The minimum number of colors required for a gap vertex-distinguishing edge coloring of G is called the gap chromatic number of G and is denoted by gap(G). In this paper, we study the gap chromatic number for a large set of graphs G of order n and we even prove that gap(G) ∈ {n− 1, n, n + 1}.