Rainbow directed path conjecture for regular digraphs

Determine whether every d-regular digraph with a proper edge-colouring using d colours contains a directed rainbow path with d−1 edges.

Background

A directed analogue of Schrijver’s rainbow-path problem is formulated for d-regular digraphs in which the d colours form a proper edge-colouring. A positive answer would imply Graham’s conjecture, the Alspach–Liversidge conjecture, and their non-abelian generalization through coloured Cayley digraphs.

The paper proves only an asymptotic result in the dense regime, yielding paths of length (1−o(1))d(1-o(1))d, so the exact d−1-edge assertion is left unresolved.

References

Specifically, we ask the following, which may be considered to be a directed generalisation of Schrijver's problem to directed d-regular digraphs, that is, digraphs in which every vertex has in-degree and out-degree exactly d. Let G be a d-regular digraph properly edge-coloured with d colours. Does G contain a directed rainbow path with d-1 edges?

— Towards Graham's rearrangement conjecture via rainbow paths  (2503.01825 - Bucić et al., 3 Mar 2025) in Problem 1 (labelled Problem \ref{problem:directed}), Section 1, Introduction