Rainbow tight Hamilton cycles under weak global colour bounds

Determine whether a weak global bound of order o(n^{k-1}) on every colour class in a k-uniform Dirac hypergraph suffices to force a rainbow tight Hamilton cycle, or whether stronger global or local colouring bounds are necessary.

Background

The paper proves the rainbow Dirac theorem only for loose Hamilton cycles. For tight Hamilton cycles, the cited literature provides results in complete hypergraphs under a substantially stronger o(n) bound on the number of edges of each colour, while the corresponding Dirac-hypergraph problem is unresolved.

The authors explain that their switching method does not readily extend to tight cycles because tight Hamilton cycles are more connected and offer less flexibility for local alterations. They also give a 3-uniform construction showing that, for part of the minimum-codegree range above the tight Hamilton-cycle threshold, an o(n2) global bound alone does not guarantee a rainbow tight Hamilton cycle.

References

Alternatively, it could be that a weak global bound of $o(n{k-1})$ edges of each colour (as in Theorem~\ref{thm:main}) already suffices to force a rainbow tight Hamilton cycle. This remains open but the following example shows that, at least in some range of minimum degree, bounding the colouring further than the weak global bound is necessary.

— A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs  (2501.07644 - Kathapurkar et al., 13 Jan 2025) in Section 6, “Concluding Remarks”