Optimal linear order coefficient for prescribed \(3q\)-cycles

Determine the smallest constant \(\gamma\) such that an order hypothesis of the form \(n\ge\gamma q+O(1)\), together with minimum semidegree at least \(\lceil n/3\rceil\), guarantees that every vertex of an oriented graph lies on a directed cycle of length \(3q\).

Background

The main theorem proves the prescribed-vertex conclusion for directed $3q$-cycles under the explicit order bound $n\ge n_0(q)$, where for $q\ge4$ the bound is $n\ge45q-8$. The paper explains that its terminal-safe-chain argument cannot lower this bound to $45q-9$ using only the order and minimum semidegree of the remaining graph, because the resulting parameters lie on the sharp boundary of the short-linking construction.

Consequently, the paper leaves open the optimization of the asymptotic coefficient in a bound of the form $n\ge\gamma q+O(1)$.