Sufficient conditions for a digraph to contain: a pre-Hamiltonian cycle and cycles of lengths 3 and 4

Let $D$ be a digraph of order $p\ge 5$ with minimum degree at least $p-1$ and with minimum semi-degree at least $p\mathrm{/}2-1$. In his excellent and renowned paper, ``Long Cycles in Digraphs" (Proc. London Mathematical Society (3), 42 (1981), Thomassen fully characterized the following for $p=2n+1$: (i) $D$ has a cycle of length at least $2n$; and (ii) $D$ is Hamiltonian. Motivated by this result, and building on some of the ideas in Thomassen’s paper, we investigated the Hamiltonicity (when $p$ is even) and pancyclcity (when $p$ is arbitrary) such digraphs. We have given a complete description of whether such digraphs are Hamiltonian ($p$ is even), are pancyclic ($p$ is arbitrary). Since the proof is very long, we have divided it into three parts. In this paper, we provide a full description of the following: (iii) for $k=3$ and $k=4$, the digraph $D$ contains a cycle of length $k$; and (iv) the digraph $D$ contains a pre-Hamiltonian cycle, i.e. a cycle of length $p-1$.