Gap vertex-distinguishing edge colorings of graphs

Abstract

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 $G$ 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 $\mathrm{gap}\left(G\right)$.

We here study the gap chromatic number for a large set of graphs $G$ of order $n$ and prove that $\mathrm{gap}\left(G\right)\in \{n-1,n,n+1\}$.