Transversal Dirac theorem for oriented graph collections

Determine whether every collection of oriented graphs \(\mathcal{D}=\{D_1,\ldots,D_n\}\) on a common vertex set of size \(n\), with minimum semi-degree \(\delta^0(\mathcal{D})\geq \lceil(3n-4)/8\rceil\), contains a transversal directed Hamilton cycle.

Background

The paper proves a transversal analogue of Ghouila-Houri’s theorem: for sufficiently large $n$, a collection of digraphs with common vertex set of size $n$ and minimum semi-degree at least $n/2$ contains a transversal directed Hamilton cycle. The concluding remarks ask whether an analogous result holds for oriented graphs, which are digraphs with no directed 2-cycles.

The proposed threshold $\lceil(3n-4)/8\rceil$ comes from the oriented version of Dirac’s theorem proved by Keevash, Kühn, and Osthus, stating that every sufficiently large oriented graph with minimum semi-degree at least $(3n-4)/8$ contains a directed Hamilton cycle. The authors note that their absorption method does not directly extend to this lower semi-degree regime because the required absorbing structure is unavailable, and that new stable conditions and ideas would be needed.