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

They later proposed the following conjecture. If $G$ is a connected decomposable graph that is not the bow-tie, then $(G) \le 3$.

— An 18-colour bound for locally irregular decompositions  (2609.09355 - Lintzmayer et al., 8 Sep 2026) in Introduction, Conjecture 1

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