Competition hypergraphs of digraphs with certain properties II. Hamiltonicity

Martin Sonntag; Hanns-Martin Teichert

Displaying similar documents to “Competition hypergraphs of digraphs with certain properties II. Hamiltonicity”

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,...

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...

Cyclically k-partite digraphs and k-kernels

Hortensia Galeana-Sánchez, César Hernández-Cruz (2011)

Discussiones Mathematicae Graph Theory

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Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k,l)-kernel N of D is a k-independent set of vertices (if u,v ∈ N then d(u,v) ≥ k) and l-absorbent (if u ∈ V(D)-N then there exists v ∈ N such that d(u,v) ≤ l). A k-kernel is a (k,k-1)-kernel. A digraph D is cyclically k-partite if there exists a partition ${V_{i}}_{i = 0}^{k - 1}$ of V(D) such that every arc in D is a $V_{i} V_{i + 1} - a r c$ (mod k). We give a characterization for an unilateral digraph to be cyclically k-partite through...

Strong asymmetric digraphs with prescribed interior and annulus

Steven J. Winters (2001)

Czechoslovak Mathematical Journal

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The directed distance $d (u, v)$ from $u$ to $v$ in a strong digraph $D$ is the length of a shortest $u - v$ path in $D$ . The eccentricity $e (v)$ of a vertex $v$ in $D$ is the directed distance from $v$ to a vertex furthest from $v$ in $D$ . The center and periphery of a strong digraph are two well known subdigraphs induced by those vertices of minimum and maximum eccentricities, respectively. We introduce the interior and annulus of a digraph which are two induced subdigraphs involving the remaining vertices. Several results...

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...

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 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 $K_{n / 2, n / 2}$ .

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 <...

Problems remaining NP-complete for sparse or dense graphs

Ingo Schiermeyer (1995)

Discussiones Mathematicae Graph Theory

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For each fixed pair α,c > 0 let INDEPENDENT SET ( $m \leq c n^{α}$ ) and INDEPENDENT SET ( $m \geq (ⁿ ₂) - c n^{α}$ ) be the problem INDEPENDENT SET restricted to graphs on n vertices with $m \leq c n^{α}$ or $m \geq (ⁿ ₂) - c n^{α}$ edges, respectively. Analogously, HAMILTONIAN CIRCUIT ( $m \leq n + c n^{α}$ ) and HAMILTONIAN PATH ( $m \leq n + c n^{α}$ ) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with $m \leq n + c n^{α}$ edges. For each ϵ > 0 let HAMILTONIAN CIRCUIT (m ≥ (1 - ϵ)(ⁿ₂)) and HAMILTONIAN PATH (m ≥ (1 - ϵ)(ⁿ₂)) be the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH...

Kernels in the closure of coloured digraphs

Hortensia Galeana-Sánchez, José de Jesús García-Ruvalcaba (2000)

Discussiones Mathematicae Graph Theory

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Let D be a digraph with V(D) and A(D) the sets of vertices and arcs of D, respectively. A kernel of D is a set I ⊂ V(D) such that no arc of D joins two vertices of I and for each x ∈ V(D)∖I there is a vertex y ∈ I such that (x,y) ∈ A(D). A digraph is kernel-perfect if every non-empty induced subdigraph of D has a kernel. If D is edge coloured, we define the closure ξ(D) of D the multidigraph with V(ξ(D)) = V(D) and $A (ξ (D)) = ⋃_{i} (u, v) w i t h c o l o u r i t h e r e e x i s t s a m o n o c h r o m a t i c p a t h o f c o l o u r i f r o m t h e v e r t e x u t o t h e v e r t e x v c o n t a i n e d i n D .$ Let T₃ and C₃ denote the transitive tournament of order 3 and the 3-cycle,...

Hamiltonian-colored powers of strong digraphs

Garry Johns, Ryan Jones, Kyle Kolasinski, Ping Zhang (2012)

Discussiones Mathematicae Graph Theory

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For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^{k}$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^{k}$ if the directed distance $^{\to} d_{D} (u, v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^{k}$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^{k}$ is distance-colored if each arc (u, v) of $D^{k}$ is assigned the color i where $i =^{\to} d_{D} (u, v)$ . The digraph...

Nonderogatory unicyclic digraphs.

Bravo, Diego, Rada, Juan (2007)

International Journal of Mathematics and Mathematical Sciences

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On Hamiltonian consecutive-d digraphs

D. Z. Du, D. F. Hsu (1989)

Banach Center Publications

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Monochromatic cycles and monochromatic paths in arc-colored digraphs

Hortensia Galeana-Sánchez, Guadalupe Gaytán-Gómez, Rocío Rojas-Monroy (2011)

Discussiones Mathematicae Graph Theory

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We call the digraph D an m-colored digraph if the arcs of D are colored with m colors. A path (or a cycle) is called monochromatic if all of its arcs are colored alike. A cycle is called a quasi-monochromatic cycle if with at most one exception all of its arcs are colored alike. A subdigraph H in D is called rainbow if all its arcs have different colors. A set N ⊆ V(D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different...

Forbidden triples implying Hamiltonicity: for all graphs

Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson (2004)

Discussiones Mathematicae Graph Theory

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In [2], Brousek characterizes all triples of graphs, G₁, G₂, G₃, with $G_{i} = K_{1, 3}$ for some i = 1, 2, or 3, such that all G₁G₂G₃-free graphs contain a hamiltonian cycle. In [6], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁, G₂, G₃, none of which is a $K_{1, s}$ , s ≥ 3 such that G₁, G₂, G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In this paper, a characterization will be given of all triples G₁, G₂, G₃ with none being $K_{1, 3}$ , such that all...