Unrooted directed-cycle semidegree threshold

Determine whether, for every integer \(\ell\ge4\) and every integer \(k>2\) that is the smallest integer not dividing \(\ell\), every sufficiently large oriented graph on \(n\) vertices with minimum semidegree at least \(\lfloor n/k\rfloor+1\) contains a directed cycle of length \(\ell\).

Background

The paper distinguishes the prescribed-vertex threshold studied in its main theorem from the ordinary unrooted containment problem, where the cycle need only occur somewhere in the oriented graph. It reports a conjecture of Kelly, Kühn and Osthus asserting that the smallest integer $k>2$ not dividing the target cycle length $\ell$ determines the asymptotic semidegree threshold $\lfloor n/k\rfloor+1$.

The cited conjecture is motivated by cyclic blow-up constructions. The paper notes that subsequent work establishes corresponding asymptotic results in some parameter ranges and that later work determines corrected exact thresholds for the unrooted problem, but the quoted conjecture is explicitly presented as a conjecture and is therefore included as an unresolved question stated in the paper.