The Prescribed-Vertex Semidegree Threshold for Directed $3q$-Cycles in Oriented Graphs

Abstract

For every $q\ge 2$, we prove that every oriented graph $G$ on $n\ge 45q-8$ vertices whose minimum semidegree satisfies \[ δ^0(G)\ge \left\lceil\frac n3\right\rceil \] contains a directed cycle of length $3q$ through every vertex. The semidegree bound is sharp. This closes the one-unit gap left by the prescribed-vertex theorem of Kelly, Kühn and Osthus when $3\mid n$. We also prove that if an oriented graph $H$ has order $N$, minimum semidegree $d\ge 3$, and $7d\ge 2N+3$, then every ordered pair of distinct vertices is joined by a path of length three, four, or five. The constant $+3$ is best possible. As a consequence, the order hypothesis $n\ge 10^{10}\ell$ in the general prescribed-vertex theorem of Kelly, Kühn and Osthus can be replaced by $n\ge 15\ell -60$ for $\ell \ge 7$.