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arXiv · 2510.26273

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

Samvel Kh. Darbinyan
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Abstract

Let $D$ be a digraph of order $p\geq5$ with minimum degree at least $p-1$ and with minimum semi-degree at least $p/2-1$. 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 $p-1$.

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Samvel Kh. Darbinyan. 2025-10-30. Sufficient conditions for a digraph to contain: a pre-Hamiltonian cycle and cycles of lengths 3 and 4. https://arxiv.org/abs/2510.26273

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