Rainbow paths in Eulerian digraphs of prescribed average degree

Determine the maximum length of a rainbow path guaranteed in every properly edge-coloured Eulerian digraph of average out-degree d.

Background

The unrestricted directed analogue of the rainbow Turán problem is too weak because directed bicliques can have large average out-degree while containing no directed path of length two.

The paper proposes Eulerian digraphs—digraphs whose in-degree equals out-degree at every vertex—as a natural setting in which to seek a degree-dependent lower bound for rainbow path length. The corresponding uncoloured problem is itself linked to an unresolved conjecture of Bollobás and Scott.