On the Vertex Distinguishing Equitable Edge-Coloring of Graphs

Abstract

A vertex-distinguishing edge-coloring (VDEC) of a simple graph \(G\) which contains no more than one isolated vertex and no isolated edge is equitable (VDEEC) if the absolute value of the difference between the number of edges colored by color \(i\) and the number of edges colored by color \(j\) is at most one. The minimal number of colors needed such that \(G\) has a VDEEC is called the vertex-distinguishing equitable chromatic index of \(G\). In this paper, we propose two conjectures after investigating VDEECs on some special families of graphs, such as the stars, fans, wheels, complete graphs, complete bipartite graphs, etc.