Proper cycles of indefinite quadratic forms and their right neighbors

Ahmet Tekcan

Displaying similar documents to “Proper cycles of indefinite quadratic forms and their right neighbors”

Cycle-pancyclism in bipartite tournaments I

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ [(n+4)/2], there exists a directed cycle $C_{h (k)}$ of length h(k), h(k) ∈ k,k-2 with $| A (C_{h (k)}) \cap A (γ) | \geq h (k) - 3$ and the result is best possible. In a forthcoming paper the case of directed cycles of length k, k even and k <...

Cycle-pancyclism in bipartite tournaments II

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T[V(γ)]? It is proved that for an even k in the range (n+6)/2 ≤ k ≤ n-2, there exists a directed cycle $C_{h (k)}$ of length h(k), h(k) ∈ k,k-2 with $| A (C_{h (k)}) \cap A (γ) | \geq h (k) - 4$ and the result is best possible. In a previous paper a similar result for 4 ≤ k ≤ (n+4)/2 was...

A conjecture on cycle-pancyclism in tournaments

Hortensia Galeana-Sánchez, Sergio Rajsbaum (1998)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian tournament with n vertices and γ a hamiltonian cycle of T. In previous works we introduced and studied the concept of cycle-pancyclism to capture the following question: What is the maximum intersection with γ of a cycle of length k? More precisely, for a cycle Cₖ of length k in T we denote $I_{γ} (C ₖ) = | A (γ) \cap A (C ₖ) |$ , the number of arcs that γ and Cₖ have in common. Let $f (k, T, γ) = m a x I_{γ} (C ₖ) | C ₖ \subset T$ and f(n,k) = minf(k,T,γ)|T is a hamiltonian tournament with n vertices, and γ a hamiltonian cycle of T. In previous...

An upper bound on the basis number of the powers of the complete graphs

Salar Y. Alsardary (2001)

Czechoslovak Mathematical Journal

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The basis number of a graph $G$ is defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$ -fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is $\leq 2$ . Schmeichel proved that the basis number of the complete graph $K_{n}$ is at most $3$ . We generalize the result of Schmeichel by showing that the basis number of the $d$ -th power of $K_{n}$ is at most $2 d + 1$ .

On the limit cycle of the Liénard equation

Kenzi Odani (2000)

Archivum Mathematicum

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In the paper, we give an existence theorem of periodic solution for Liénard equation $\dot{x} = y - F (x)$ , $\dot{y} = - g (x)$ . As a result, we estimate the amplitude $ρ (μ)$ (maximal $x$ -value) of the limit cycle of the van der Pol equation $\dot{x} = y - μ (x^{3} / 3 - x)$ , $\dot{y} = - x$ from above by $ρ (μ) < 2.3439$ for every $μ \neq 0$ . The result is an improvement of the author’s previous estimation $ρ (μ) < 2.5425$ .