## Samvel Kh. Darbinyan - A new sufficient condition for a 2-strong digraph to be Hamiltonian

dmtcs:11560 - Discrete Mathematics & Theoretical Computer Science, June 24, 2024, vol. 26:2 - https://doi.org/10.46298/dmtcs.11560
A new sufficient condition for a 2-strong digraph to be HamiltonianArticle

Authors: Samvel Kh. Darbinyan

In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian: {\it Let $D$ be a 2-strong digraph of order $n\geq 9$. If $n-1$ vertices of $D$ have degrees at least $n+k$ and the remaining vertex has degree at least $n-k-4$, where $k$ is a non-negative integer, then $D$ is Hamiltonian}. This is an extension of Ghouila-Houri's theorem for 2-strong digraphs and is a generalization of an early result of the author (DAN Arm. SSR (91(2):6-8, 1990). The obtained result is best possible in the sense that for $k=0$ there is a digraph of order $n=8$ (respectively, $n=9$) with the minimum degree $n-4=4$ (respectively, with the minimum $n-5=4$) whose $n-1$ vertices have degrees at least $n-1$, but it is not Hamiltonian. We also give a new sufficient condition for a 3-strong digraph to be Hamiltonian-connected.

Volume: vol. 26:2
Section: Graph Theory
Published on: June 24, 2024
Accepted on: March 25, 2024
Submitted on: July 10, 2023
Keywords: Mathematics - Combinatorics