Rainbow walks in properly coloured regular digraphs

Determine whether every d-regular digraph properly edge-coloured with d colours contains a rainbow walk having d−o(d) distinct vertices.

Background

The exact directed analogue of Schrijver’s rainbow-path problem asks for a rainbow directed path with d−1 edges, but the paper’s methods do not resolve that question for general properly coloured regular digraphs.

The stated relaxation permits a walk rather than a path and asks only that the walk contain d−o(d) distinct vertices. The authors identify it as a more approachable open problem and note that their general rainbow-walk theorem gives a positive answer in the special case of coloured Cayley graphs.

References

We pose the following relaxation as a more approachable open problem, with Theorem~\ref{thm:weakasymptotic-intro} already giving a positive answer for coloured Cayley graphs. Let G be a d-regular digraph properly edge-coloured with d colours. Does G contain a rainbow walk with d-o(d) distinct vertices?

— Towards Graham's rearrangement conjecture via rainbow paths  (2503.01825 - Bucić et al., 3 Mar 2025) in Problem 2, Section 7 (Concluding remarks), subsection “Rainbow walks in regular digraphs”

Regarding general coloured d-regular digraphs, our current methods appear to be too weak to give a positive answer to \Cref{problem:directed}. We pose the following relaxation as a more approachable open problem, with Theorem~\ref{thm:weakasymptotic-intro} already giving a positive answer for coloured Cayley graphs. Let G be a d-regular digraph properly edge-coloured with d colours. Does G contain a rainbow walk with d-o(d) distinct vertices?

— Towards Graham's rearrangement conjecture via rainbow paths  (2503.01825 - Bucić et al., 3 Mar 2025) in Problem \ref{problem:directedwalks}, Section 6, Concluding remarks