# A new sufficient condition for a Digraph to be Hamiltonian-A proof of Manoussakis Conjecture

## TL;DR

This paper proves Manoussakis's conjecture that a 2-strongly connected digraph with a specific degree sum condition is Hamiltonian, and additionally shows such digraphs contain cycles of all lengths under certain degree constraints.

## Contribution

It confirms Manoussakis's conjecture and establishes the existence of cycles of all lengths in certain degree-satisfying digraphs.

## Key findings

- Proves the conjecture that degree sum conditions imply Hamiltonicity.
- Shows existence of cycles of all lengths in specific digraphs.
- Provides a new sufficient condition for Hamiltonian digraphs.

## Abstract

Y. Manoussakis (J. Graph Theory 16, 1992, 51-59) proposed the following conjecture. \noindent\textbf{Conjecture}. {\it Let $D$ be a 2-strongly connected digraph of order $n$ such that for all distinct pairs of non-adjacent vertices $x$, $y$ and $w$, $z$, we have $d(x)+d(y)+d(w)+d(z)\geq 4n-3$. Then $D$ is Hamiltonian.} In this paper, we confirm this conjecture. Moreover, we prove that if a digraph $D$ satisfies the conditions of this conjecture and has a pair of non-adjacent vertices $\{x,y\}$ such that $d(x)+d(y)\leq 2n-4$, then $D$ contains cycles of all lengths $3, 4, \ldots , n$.