Minimum total-degree threshold for factors in sparse-independence digraphs

Determine the minimum total-degree condition that guarantees an H-factor in every sufficiently large n-vertex digraph D with independence number (D)=o(n), for each digraph H whose order divides n.

Background

The paper places this question in the context of RamseyTuran factor problems for digraphs. Earlier results had established thresholds for particular factors, including transitive tournament factors, but a general degree criterion for an arbitrary digraph H remained unresolved.

The question asks for an optimal or otherwise sufficient minimum total-degree threshold under the structural assumption that the host digraph has sublinear independence number. It is broader than the cycle-factor problem studied in the paper and is presented as a question posed in prior work rather than solved by the results here.

References

Subsequently, Wang, Wang and Yan determined the optimal threshold for $TT_3$-factors and posed the following question. Let $D$ be an $n$-vertex digraph with $\alpha(D)=o(n)$, and let $H$ be a digraph satisfying $|V(H)|\mid n$. What minimum total-degree condition on $D$ guarantees that $D$ contains an $H$-factor?

— Dominant-Degree Conditions for Ramsey--Turán Factors of Non-Directed Cycle Orientations  (2609.00889 - Zhou et al., 1 Sep 2026) in Section 1, Introduction, Question 1 (attributed to Wang, Wang and Yan)