Three-color conjecture for (+,+)-locally irregular colorings

Prove that every oriented graph has (+,+)-locally irregular chromatic index at most three.

Background

The paper describes the (+,+)-local irregularity notion, in which adjacent vertices are distinguished by their outdegrees within each color class. Bensmail and Renault established constant upper bounds and NP-completeness results for this parameter, but the stated three-color bound remains a conjecture.

References

Bensmail and Renault in observed that there are no digraphs which are non $(+,+)$-locally irregularly colorable and proposed the following conjecture.

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

Bensmail et al. observed that only digraphs which contain (an oriented) sink-source path do not have $(+,-)$-locally irregular coloring and proposed the following conjecture.

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