The four-color bound for locally irregular edge colorings of colorable graphs

All graphs in this paper are finite and simple. A graph is locally irregular if the degrees of the endpoints of every edge are distinct. A locally irregular edge coloring is an edge coloring in which every color class induces a locally irregular subgraph. A graph is colorable if it admits such a coloring, and for a colorable graph $G$, let $\chi_{\operatorname{irr}}^{\prime}(G)$ be the minimum number of colors in a locally irregular edge coloring.

Four-color bound. Every colorable connected graph $G$ satisfies

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

This is presented as a weaker version of the refuted Local Irregularity Conjecture, motivated by the bow-tie graph, which requires four colors. The supplied text does not establish whether the four-color bound is proved or remains open.