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Zh. G. Nikoghosyan

Publications and source records attributed to Zh. G. Nikoghosyan.

At least 19 recordsLinked to original sources
arXiv2016

On Dirac's Conjecture

Let $G$ be a 2-connected graph, $l$ be the length of a longest path in $G$ and $c$ be the circumference - the length of a longest cycle in $G$. In 1952, Dirac proved that $c>\sqrt{2l}$ and conjectured that $c\ge 2\sqrt{l}$. In this paper we present more general sharp bounds in terms of $l$ and the length $m$ of a vine on a longest path in $G$ including Dirac's conjecture as a corollary: if $c=m+y+2$ (generally, $c\ge m+y+2$) for some integer $y\ge 0$, then $c\ge\sqrt{4l+(y+1)^2}$ if $m$ is odd; and $c\ge\sqrt{4l+(y+1)^2-1}$ if $m$ is even.

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arXiv2015

On large $k$-ended trees in connected graphs

A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let $σ_m$ be the minimum degree sum of an independent set of $m$ vertices. The main result (Theorem 2) provides a lower bound for $t_{k+1}$ in terms of $σ_m$ and relative orders: if $G$ is a connected graph and $k$, $λ$, $m$ are positive integers with $2\le m\le\min\{k,λ\}+1$ then either $t_{k+1} \ge σ_m +λ(k-m+1)+1$ or $t_k\ge t_{k+1}-λ+1$.

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arXiv2014

On $k$-ended spanning and dominating trees

A tree with at most $k$ leaves is called a $k$-ended tree. A spanning 2-ended tree is a Hamilton path. A Hamilton cycle can be considered as a spanning 1-ended tree. The earliest result concerning spanning trees with few leaves states that if $k$ is a positive integer and $G$ is a connected graph of order $n$ with $d(x)+d(y)\ge n-k+1$ for each pair of nonadjacent vertices $x,y$, then $G$ has a spanning $k$-ended tree. In this paper, we improve this result in two ways, and an analogous result is proved for dominating $k$-ended trees based on the generalized parameter $t_k$ - the order of a largest $k$-ended tree. In particular, $t_1$ is the circumference (the length of a longest cycle), and $t_2$ is the order of a longest path.

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arXiv2014

On Relative Length of Long Paths and Cycles in Graphs

Let $G$ be a graph on $n$ vertices, $p$ the order of a longest path and $κ$ the connectivity of $G$. In 1989, Bauer, Broersma Li and Veldman proved that if $G$ is a 2-connected graph with $d(x)+d(y)+d(z)\ge n+κ$ for all triples $x,y,z$ of independent vertices, then $G$ is hamiltonian. In this paper we improve this result by reducing the lower bound $n+κ$ to $p+κ$.

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arXiv2014

A Common Generalization of Dirac's two Theorems

Let $G$ be a 2-connected graph of order $n$ and let $c$ be the circumference - the order of a longest cycle in $G$. In this paper we present a sharp lower bound for the circumference based on minimum degree $δ$ and $p$ - the order of a longest path in $G$. This is a common generalization of two earlier classical results for 2-connected graphs due to Dirac: (i) $c\ge \min\{n,2δ\}$; and (ii) $c\ge\sqrt{2p}$. Moreover, the result is stronger than (ii).

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arXiv2014

Relative Length of Long Paths and Cycles in Graphs

For a graph $G$, $n$ denotes the order of $G$, $p$ the order of a longest path in $G$ and $c$ the order of a longest cycle. We show that if $G$ is a 2-connected graph such that $d(x)+d(y)+d(z)\ge p+2$ for all triples $x,y,z$ of independent vertices, then $c\ge p-1$. This improves results of Nash-Williams (in terms of minimum degree $δ$ and order $n$), Bondy (in terms of degree sum $σ_{3}$ and order $n$), and Enomoto, Heuvel, Kaneko and Saito (in terms of degree sum $σ_3$, order $n$ and relative length $diff(G)=p-c$).

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arXiv2014

On some Versions of Conjectures of Bondy and Jung

Using algebraic transformations and equivalent reformulations we derive a number of new results from some earlier ones (by the author) in more accepted terms closely related to well-known conjectures of Bondy and Jung including a number of classical results in hamiltonian graph theory (due to Dirac, Ore, Nash-Williams, Bondy, Jung and so on) as special cases. A number of extended and strengthened versions of these conjectures are proposed.

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arXiv2014

Long Cycles in 1-tough Graphs

In 1952, Dirac proved that every 2-connected graph with minimum degree $δ$ either is hamiltonian or contains a cycle of length at least $2δ$. In 1986, Bauer and Schmeichel enlarged the bound $2δ$ to $2δ+2$ under additional 1-tough condition - an alternative and more natural necessary condition for a graph to be hamiltonian. In fact, the bound $2δ+2$ is sharp for a graph on $n$ vertices when $n\equiv 1(mod\ 3)$. In this paper we present the final version of this result which is sharp for each $n$: every 1-tough graph either is hamiltonian or contains a cycle of length at least $2δ+2$ when $n\equiv 1(mod\ 3)$, at least $2δ+3$ when $n\equiv 2(mod\ 3)$ or $n\equiv 1(mod\ 4)$, and at least $2δ+4$ otherwise.

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arXiv2012

Evolutionary Hamiltonian Graph Theory

We present an alternative domain concerning mathematics to investigate universal evolution mechanisms by focusing on large cycles theory (LCT) - a simplified version of well-known hamiltonian graph theory. LCT joins together a number of $NP$-complete cycle problems in graph theory. $NP$-completeness is the kay factor insuring (by conjecture of Cook) the generation of endless developments and great diversity around large cycles problems. Originated about 60 years ago, the individuals (claims, propositions, lemmas, conjectures, theorems, and so on) in LCT continually evolve and adapt to their environment by an iterative process from primitive beginnings to best possible theorems based on inductive reasoning. LCT evolves much more rapidly than biosphere and has a few thousand pronounced species (theorems). Recall that life on earth with more than 2 million species was originated about 3.7 billion years ago and evolves extremely slowly. We show that all theorems in LCT have descended from some common primitive propositions such as "every complete graph is hamiltonian" or "every graph contains a cycle of length at least one" via improvements, modifications and three kinds of generalizations - closing, associating and extending. It is reasonable to review Darwinian mechanisms in light of LCT evolution mechanisms (especially inductive reasoning) including the origin and macroevolution disputable phenomena in the biosphere.

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arXiv2012

A Note on Hamilton Cycles

If $G$ is a more than one tough graph on $n$ vertices with $δ\ge \frac{n}{2}-a$ for a given $a>0$ and $n$ is large enough then $G$ is hamiltonian.

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arXiv2012

Nonhamiltonian Graphs with Given Toughness

In 1973, Chvátal introduced the concept of toughness $τ$ of a graph and constructed an infinite class of nonhamiltonian graphs with $τ=3/2$. Later Thomassen found nonhamiltonian graphs with $τ>3/2$, and Enomoto et al. constructed nonhamiltonian graphs with $τ=2-ε$ for each positive $ε$. The last result in this direction is due to Bauer, Broersma and Veldman, which states that for each positive $ε$, there exists a nonhamiltonian graph with $τ\ge 9/4-ε$. In this paper we prove that for each rational number $t$ with $0<t<9/4$, there exists a nonhamiltonian graph with $τ=t$.

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arXiv2012

Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles

In 1974, Goodman and Hedetniemi proved that every 2-connected $(K_{1,3},K_{1,3}+e)$-free graph is hamiltonian. This result gave rise many other hamiltonicity conditions for various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In 1997, it was proved that a single forbidden connected subgraph $R$ in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with $R=P_3$. In this paper we prove that a single forbidden subgraph $R$ can create a non trivial class of hamiltonian graphs if $R$ is disconnected: $(\ast1)$ every $(K_1\cup P_2)$-free graph either is hamiltonian or belongs to a well defined class of non hamiltonian graphs; $(\ast2)$ every 1-tough $(K_1\cup P_3)$-free graph is hamiltonian. We conjecure that every 1-tough $(K_1\cup P_4)$-free graph is hamiltonian and every 1-tough $P_4$-free graph is hamiltonian

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arXiv2012

My Research Visiting Card in Hamiltonian Graph Theory

We present eighteen exact analogs of six well-known fundamental Theorems (due to Dirac, Nash-Williams and Jung) in hamiltonian graph theory providing alternative compositions of graph invariants. In Theorems 1-3 we give three lower bounds for the length of a longest cycle $C$ of a graph $G$ in terms of minimum degree $δ$, connectivity $κ$ and parameters $\bar{p}$, $\bar{c}$ - the lengths of a longest path and longest cycle in $G\backslash C$, respectively. These bounds have no analogs in the area involving $\bar{p}$ and $\bar{c}$ as parameters. In Theorems 11 and 12 we give two Dirac-type results for generalized cycles including a number of fundamental results (concerning Hamilton and dominating cycles) as special cases. Connectivity invariant $κ$ appears as a parameter in some fundamental results and in some their exact analogs (Theorems 3-10) in the following chronological order: 1972 (Chvátal and Erdös), 1981a (Nikoghosyan), 1981b (Nikoghosyan), 1985a (Nikoghosyan), 1985b (Nikoghosyan), 2000 (Nikoghosyan), 2005 (Lu, Liu, Tian), 2009 (Nikoghosyan), 2009a (Yamashita), 2009b (Yamashita), 2011a (Nikoghosyan), 2011b (Nikoghosyan).

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arXiv2012

Degree Conditions for Dominating Cycles in 1-tough Graphs

We prove: (i) if $G$ is a 1-tough graph of order $n$ and minimum degree $δ$ with $δ\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle unless $G$ belongs to an easily specified class of graphs with $κ(G)=2$ and $τ(G)=1$. The second result follows immediately from the first result: (ii) if $G$ is a 3-connected 1-tough graph with $δ\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.

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arXiv2011

A Size Upper Bound for Dominating Cycles

Recently it was shown (by the author) that every graph of size $q$ (the number of edges) and minimum degree $δ$ is hamiltonian if $q\leδ^2+δ-1$ (arXiv:1107.2201v1). In this paper we present the exact analog of this result for dominating cycles: if $G$ is a 2-connected graph with $q\le8$ if $δ=2$ and $q\le (3(δ-1)(δ+2)-1)/2$ if $δ\ge3$, then each longest cycle in $G$ is a dominating cycle. The result is sharp in all respects.

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