The four-color bound for locally irregular edge colorings

A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring is locally irregular if every color induces a locally irregular subgraph. A graph is colorable if it admits a locally irregular edge coloring. For a colorable graph $G$, let $\chi_{\rm irr}^{\prime}(G)$ be the smallest number of colors required by such a coloring. The four-color bound. Every colorable connected graph $G$ satisfies

$$\chi_{\rm irr}^{\prime}(G)\leq 4.$$

This weaker version of the Local Irregularity Conjecture is established for all colorable cacti, as stated in the paper’s context, so the candidate is treated as solved.