New bounds for locally irregular chromatic index of bipartite and subcubic graphs

TL;DR

This paper improves bounds on the number of colors needed for locally irregular edge-colorings in bipartite and subcubic graphs, reducing previous upper bounds significantly and establishing new results.

Contribution

The paper provides new upper bounds of 7 and 220 colors for bipartite and general graphs respectively, and proves 4 colors suffice for subcubic graphs.

Findings

Bipartite graphs require at most 7 colors for locally irregular edge-coloring.

General graphs require at most 220 colors for locally irregular edge-coloring.

Subcubic graphs can be colored with at most 4 colors for local irregularity.

Abstract

A graph is \textit{locally irregular} if the neighbors of every vertex $v$ have degrees distinct from the degree of $v$. \textit{locally irregular edge-coloring} of a graph $G$ is an (improper) edge-coloring such that the graph induced on the edges of any color class is locally irregular. It is conjectured that $3$ colors suffice for a locally irregular edge-coloring. Recently, Bensmail et al. (Bensmail, Merker, Thomassen: Decomposing graphs into a constant number of locally irregular subgraphs, {\em European J. Combin.}, 60:124--134, 2017) settled the first constant upper bound for the problem to $328$ colors. In this paper, using a combination of existing results, we present an improvement of the bounds for bipartite graphs and general graphs, setting the best upper bounds to $7$ and $220$, respectively. In addition, we also prove that $4$ colors suffice for locally irregular edge-coloring of any subcubic graph.