An improved upper bound on the adjacent vertex distinguishing chromatic index of a graph

TL;DR

This paper establishes a tighter upper bound on the adjacent vertex distinguishing chromatic index for graphs, improving previous bounds and contributing to the understanding of edge colorings that distinguish adjacent vertices.

Contribution

It proves a new upper bound of 5(Δ+2)/2 for the adjacent vertex distinguishing chromatic index, refining earlier results that used a bound of 3Δ.

Findings

New upper bound of 5(Δ+2)/2 for the chromatic index

Improves previous bound of 3Δ for graphs without isolated edges

Applicable to graphs with maximum degree Δ and no isolated edges

Abstract

An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing coloring of G is denoted by $\chi'_a(G)$. In this paper, we prove that $\chi_a'(G)$ <= 5($\Delta+2$)/2 for any graph G having maximum degree $\Delta$ and no isolated edges. This improves a result in [S. Akbari, H. Bidkhori, N. Nosrati, r-Strong edge colorings of graphs, Discrete Math. 306 (2006), 3005-3010], which states that $\chi_a'(G)$ <= 3$\Delta$ for any graph G without isolated edges.