Sufficient conditions for a digraph to contain: a pre-Hamiltonian cycle and cycles of lengths 3 and 4
Samvel Kh. Darbinyan
TL;DR
The article establishes precise structural conditions under which a digraph $D$ with order $p\ge5$ and strong degree constraints must contain small cycles or near-Hamiltonian structures. It leverages insertion lemmas and a framework of exceptional digraph families to prove: (i) under $\delta(D)\ge p-1$ and $\delta^0(D)\ge p/2-1$, $D$ contains a $C_3$ unless it lies in specific bipartite-like families; (ii) under the same conditions, $D$ contains a $C_4$ unless it belongs to particular exceptional digraphs; and (iii) either $D$ contains a pre-Hamiltonian cycle or again falls into a union of well-characterized exceptional families. The results extend Thomassen's and related Hamiltonicity/pancyclicity theories to give complete descriptions for cycles of lengths $3$ and $4$ and for pre-Hamiltonian cycles, highlighting the role of near-Hamiltonian structures and a finite set of extremal graphs. Overall, the paper contributes a rigorous dichotomy: either the digraph exhibits the desired cycle properties, or it is one of a small, explicit list of extremal configurations, under strong degree conditions.
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 $p-1$.