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S. Kh. Darbinyan

Publications and source records attributed to S. Kh. Darbinyan.

6 recordsLinked to original sources
arXiv2018

On Hamiltonian and Hamilton-connected digraphs

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).

math.CO↗
arXiv2014

On Cycles through Vertices of Large Semidegree in Digraphs

Let $D$ be a strong digraph on $n=2m+1\geq 5$ vertices. In this paper we show that if $D$ contains a cycle of length $n-1$, then $D$ has also a cycle which contains all vertices with in-degree and out-degree at least $m$ (unless some extremal cases).

math.CO↗
arXiv2012

On longest non-Hamiltonian Cycles in Digraphs with the Conditions of Bang-Jensen, Gutin and Li

Let $D$ be a strong digraph on $n\geq 4$ vertices. In [2, J. Graph Theory 22 (2) (1996) 181-187)], J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If (*) $d(x)+d(y)\geq 2n-1$ and $min \{d(x), d(y)\}\geq n-1$ for every pair of non-adjacent vertices $x, y$ with a common in-neighbour or (**) $min \{d^+(x)+ d^-(y),d^-(x)+ d^+(y)\}\geq n$ for every pair of non-adjacent vertices $x, y$ with a common in-neighbour or a common out-neighbour, then $D$ is hamiltonian. In this paper we show that: (i) if $D$ satisfies the condition (*) and the minimum semi-degree of $D$ at least two or (ii) if $D$ is not directed cycle and satisfies the condition (**), then either $D$ contains a cycle of length $n-1$ or $n$ is even and $D$ is isomorphic to complete bipartite digraph or to complete bipartite digraph minus one arc.

math.CO↗
arXiv2012

A Note on Long non-Hamiltonian Cycles in One Class of Digraphs

Let $D$ be a strong digraph on $n\geq 4$ vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the following theorem: if (*) $d(x)+d(y)\geq 2n-1$ and $min \{d^+(x)+ d^-(y),d^-(x)+ d^+(y)\}\geq n-1$ for every pair of non-adjacent vertices $x, y$ with a common in-neighbour or a common out-neighbour, then $D$ is hamiltonian. In this note we show that: if $D$ is not directed cycle and satisfies the condition (*), then $D$ contains a cycle of length $n-1$ or $n-2$.

math.CO↗
arXiv2011

On the pancyclicity of digraphs with large semi-degrees

Let $D$ be an directed graph on $p\geq 10$ vertices with minimum degree at least $p-1$ and minimum semi-degree at least $ p/2 -1$. We present a detailed proof of the following result [13]: The digraph $D$ is pancyclic, unless some extremal cases (which are characterized).

math.CO↗