On Hamiltonian and Hamilton-connected digraphs

TL;DR

This paper disprove a conjecture about 3-strongly connected digraphs being Hamiltonian-connected under certain degree conditions, and supports a related conjecture for 4-strongly connected digraphs.

Contribution

It disproves a specific conjecture on Hamiltonian connectivity in 3-strong digraphs and proves results supporting a conjecture for 4-strong digraphs.

Findings

Disproved Conjecture 1 regarding 3-strongly connected digraphs.

Proved results supporting Conjecture 2 for 4-strongly connected digraphs.

Provided detailed proofs in English for these results.

Abstract

C. Thomassen in \cite{[11]} suggested (see also \cite{[2]}, J. C.Bermond, C. Thomassen, Cycles in Digraphs - A survey, J. Graph Theory 5 (1981) 1-43, Conjectures 1.6.7 and 1.6.8) the following conjectures : 1. Every 3-strongly connected digraph of order $n$ and with minimum degree at least $n+1$ is strongly Hamiltonian-connected. 2. Let $D$ be a 4-strongly connected digraph of order $n$ such that the sum of the degrees of any pair of non-adjacent vertices is at least $2n+1$. Then $D$ is strongly Hamiltonian-connected. We disprove Conjecture 1 and prove two results which provide some support for Conjecture 2. The main goal of this article is to present the detailed proofs of these results (in English).