An Improved Upper Bound on the Adjacent Vertex Distinguishing Edge Chromatic Number of a Simple Graph

Abstract

An adjacent vertex distinguishing edge coloring, or an avd-coloring,
of a simple graph \(G\) is a proper edge coloring of \(G\) such that
no two adjacent vertices are incident with the same set of colors.

H. Hatami showed that every simple graph \(G\) with no isolated
edges and maximum degree \(\Delta\) has an avd-coloring with at
most \(\Delta + 300\) colors, provided that \(\Delta > 10^{20}\).

We improve this bound as follows: if \(\Delta > 10^{15}\), then the
avd-chromatic number of \(G\) is at most \(\Delta + 180\), where
\(\Delta\) is the maximum degree of \(G\).