Two-color conjecture for weak local irregularity

Prove that every digraph has weak locally irregular chromatic index at most two, where adjacent vertices in each color class must have different outdegree-indegree pairs.

Background

Weak local irregularity distinguishes adjacent vertices by their ordered pairs of outdegree and indegree in each monochromatic subdigraph. The paper gives a general five-color upper bound and proves the conjectured two-color bound for several classes, including digraphs with bipartite or tripartite skeletons, tournaments, and symmetric digraphs. The general assertion remains unresolved.

References

We propose a conjecture that 2 colors are enough for a weak local irregular decomposition of any digraph; this suggest that such a weakening of $(+,+)$-irregularity results in needing fewer colors in the general case (see Conjecture 3.1 in that states that ${\rm lir}{(+,+)}(D)\leq 3$ for each digraph $D$).

— Weak and strong local irregularity of digraphs  (2502.07933 - Grzelec et al., 11 Feb 2025) in Section 2, Weak local irregularity