A Note on the Vertex-Distinguishing Proper Edge Coloring of Graphs

Abstract

The number of colors required to properly color the edges of a simple graph \(G\) such that any two vertices are incident with different sets of colors is referred to as the vertex-distinguishing edge chromatic number, denoted by \(\chi’_{vd}(G)\). This paper explores an interesting phenomenon concerning vertex-distinguishing proper edge coloring. Specifically, we prove that for every integer \(m \geq 3\), there exists a graph \(G\) of maximum degree \(m\) with \(\chi’_{vd}(G) < \chi'_{vd}(H)\), where \(H\) is a proper subgraph of \(G\).