Local Irregularity Conjecture for connected locally irregular-colorable graphs

Prove that every connected locally irregular-colorable graph other than the bow-tie graph has locally irregular chromatic index at most three.

Background

The paper recalls the Local Irregularity Conjecture for simple graphs. The conjecture concerns decomposing a locally irregular-colorable graph into at most three locally irregular subgraphs, with the bow-tie graph identified as the unique known colorable cactus requiring four colors. The conjecture remains unresolved in general and motivates analogous questions for digraphs.

References

In the situation when the $1$-$2$-$3$ Conjecture was proved, the conjecture that $\mathrm{lir}(G)\leq 3$ for every locally irregular colorable graph $G$ remains the main focus of research on this topic.

— Weak and strong local irregularity of digraphs  (2502.07933 - Grzelec et al., 11 Feb 2025) in Section 1, Introduction