Schrijver’s rainbow-path conjecture

Prove that every properly edge-coloured d-regular graph whose d colours form a proper edge-colouring contains a rainbow path with d−1 edges.

Background

Schrijver’s conjecture concerns properly d-edge-coloured d-regular graphs and asks for a rainbow path using d−1 edges, hence all but one of the available colours. It generalizes rainbow-path conjectures for complete graphs and has direct consequences for rearrangement problems in groups.

The paper proves the asymptotic bound that every such graph contains a rainbow path of length (1−o(1))d, but this does not settle the exact d−1 conclusion.

References

Schrijver asked for a far reaching generalisation of Andersen's conjecture by postulating the existence of a rainbow path of length d-1 in any properly d-edge-coloured d-regular graph G, and he verified this conjecture whenever d\leq 10.

— Towards Graham's rearrangement conjecture via rainbow paths  (2503.01825 - Bucić et al., 3 Mar 2025) in Section 1 (Introduction), paragraph beginning “Schrijver asked for a far reaching generalisation”