Gap vertex-distinguishing edge colorings of graphs

In this paper, we study a new coloring parameter of graphs called the gap vertex-distinguishing edge coloring  . It consists in an edge-coloring of a graph GG which induces a vertex distinguishing labeling of GG 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 GG is called the gap chromatic number   of GG and is denoted by gap(G).We here study the gap chromatic number for a large set of graphs GG of order nn and prove that gap(G)∈{n−1,n,n+1}.