Linear forests and ordered cycles

Guantao Chen; Ralph J. Faudree; Ronald J. Gould; Michael S. Jacobson; Linda Lesniak; Florian Pfender

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Displaying similar documents to “Linear forests and ordered cycles”

Pancyclism and small cycles in graphs

Ralph Faudree, Odile Favaron, Evelyne Flandrin, Hao Li (1996)

Discussiones Mathematicae Graph Theory

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We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u)+d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u)+d(v) ≥ n+z, where z = 0 when n is odd and z = 1 otherwise, then G contains a cycle of length m for each 3 ≤ m ≤ max (dC(u,v)+1, [(n+19)/13]), $d_{C} (u, v)$ being the distance of u and v on a hamiltonian cycle...

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...

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...

Knotted Hamiltonian cycles in spatial embeddings of complete graphs.

Blain, Paul, Bowlin, Garry, Foisy, Joel, Hendricks, Jacob, LaCombe, Jason (2007)

The New York Journal of Mathematics [electronic only]

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A note on a new condition implying pancyclism

Evelyne Flandrin, Hao Li, Antoni Marczyk, Mariusz Woźniak (2001)

Discussiones Mathematicae Graph Theory

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We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to $K_{n / 2, n / 2}$ .

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 $2 d + 1$ .

Graph invariants and large cycles: a survey.

Nikoghosyan, Zh.G. (2011)

International Journal of Mathematics and Mathematical Sciences

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Cycles with a given number of vertices from each partite set in regular multipartite tournaments

Lutz Volkmann, Stefan Winzen (2006)

Czechoslovak Mathematical Journal

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If $x$ is a vertex of a digraph $D$ , then we denote by $d^{+} (x)$ and $d^{-} (x)$ the outdegree and the indegree of $x$ , respectively. A digraph $D$ is called regular, if there is a number $p \in ℕ$ such that $d^{+} (x) = d^{-} (x) = p$ for all vertices $x$ of $D$ . A $c$ -partite tournament is an orientation of a complete $c$ -partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular,...