A Note on Hamiltonian Cycles in Digraphs with Large Degrees

TL;DR

This paper proves a condition under which a 2-strong directed graph with large degrees and a specific cycle length passing through a particular vertex is guaranteed to be Hamiltonian.

Contribution

It establishes a new degree condition involving a special vertex and cycle length that ensures Hamiltonicity in 2-strong digraphs.

Findings

Degree conditions guarantee Hamiltonian cycles in certain digraphs.

A cycle of length at least n-k-2 passing through a specific vertex implies Hamiltonicity.

The result extends previous degree-based Hamiltonian criteria.

Abstract

In this note we prove: {\it Let $D$ be a 2-strong digraph of order $n$ such that its $n-1$ vertices have degrees at least $n+k$ and the remaining vertex $z$ has degree at least $n-k-4$, where $k$ is a positive integer. If $D$ contains a cycle of length at least $n-k-2$ passing through $z$, then $D$ is Hamiltonian}.