Pancyclic in-tournaments

Tewes, Meike

doi:10.1016/s0012-365x(00)00238-7

Cited by 11 publications

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“…Thus, there is no arc from !1 to N0 and we have d N 1 )). By the same arguments we have already used in (5) to show that N 1 = ∅, we now deduce that N ! 1 = ∅.…”

Section: Resultssupporting

confidence: 63%

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Vertex 6-pancyclic in-tournaments

Petèrs

Volkmann

2004

Discrete Mathematics

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An in-tournament is a loopless digraph without multiple arcs and cycles of length 2 such that the negative neighborhood of every vertex induces a tournament. This paper tackles the problem of vertex k-pancyclicity in strong in-tournaments of order n, i.e., every vertex belongs to a cycle of length l for every k 6 l 6 n. In 2001, Tewes and Volkmann (J. Graph Theory 36 (2001) 84) gave sharp lower bounds for the minimum degree such that a strong in-tournament is vertex k-pancyclic for k 6 5 and k ¿ n − 3. In accordance with a family of examples (see J. Graph Theory 36 (2001) 84) and their results, they conjectured that every strong in-tournament of order n with minimum degree greater than 9(n−k−1) 5+6k+(−1) k 2 −k+2 + 1 is vertex k-pancylic. The main result of this paper is the conÿrmation of this conjecture for the case k = 6 (except for the values n = 14; 15; 16). Terminology and introductionThroughout this paper we will consider ÿnite digraphs without loops and multiple arcs. As usual for a digraph D its vertex set is denoted by V (D) and the arc set by E(D). For two distinct vertices u; v ∈ V (D), the notation u → v is used to indicate that there is an arc from u to v. We also write uv ∈ E(D) and say that u and v are adjacent, where u is called a negative neighbor of v and v is called a positive neighbor of u. Moreover, let D1 and D2 be two disjoint subdigraphs of D. If there is an arc from every vertex in D1 to every vertex in D2, then we say that D1 dominates D2, indicated by D1 → D2. In the case D1 = {z} (D2 = {ẑ}), we also use the short form z → D2 (D1 →ẑ).For an arbitrary vertex z ∈ V (D), we deÿne the positive neighborhood of z, N + (z) = N + (z; D), as the set of all positive neighbors of z in D. Analogously, N − (z) = N − (z; D) consists of all negative neighbors of z in D and is referred to as the negative neighborhood of z. Then we deÿne the indegree of z by d − (z) = d − (z; D) = |N − (z)| and accordingly the outdegree of z by d + (z) = d + (z; D) = |N + (z)|. More general, N + (X ) = N + (X; D) = z∈X N + (z) and N − (X ) = N − (X; D) = z∈X N − (z) for a subset X of V (D). For a subdigraph S of D, the positive and negative neighborhood of z with respect to S, N + (z; S) and N − (z; S), are given by N + (z) ∩ V (S) and N − (z) ∩ V (S), respectively. Analogously, we deÿne d + (z; S), d − (z; S), N + (X; S) and N − (X; S). Furthermore, d + (X; S) = |N + (X; S)| and d − (X; S) = |N − (X; S)|. Moreover, D[X ] denotes the subdigraph that is induced by the subset X of V (D). The minimum degree (D) of D is deÿned by (D) = min{ − (D); + (D)}, where − (D) = min z∈V (D) d − (z) and + (D) = min z∈V (D) d + (z).Whenever we refer to cycles and paths, we mean by this oriented cycles and oriented paths. A cycle C of length k is also called a k-cycle. We say that a digraph D is strongly connected or just strong, if for every pair of distinct vertices (u; v), u; v ∈ V (D), there is a path from u to v. A digraph D is connected, if the underlying graph is connected. We only consider connected digraphs in this paper.

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“…Thus, there is no arc from !1 to N0 and we have d N 1 )). By the same arguments we have already used in (5) to show that N 1 = ∅, we now deduce that N ! 1 = ∅.…”

Section: Resultssupporting

confidence: 63%

“…In [4][5][6][7][8], the authors focused on aspects dealing with the cycle structure of strong in-tournaments. In particular, they investigate the in uence of the minimum degree and the minimum indegree on the cycle structure of strong in-tournaments.…”

Section: Terminology and Introductionmentioning

confidence: 99%

Vertex 6-pancyclic in-tournaments

Petèrs

Volkmann

2004

Discrete Mathematics

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“…In 1993, Bang-Jensen et al [3] introduced the family of in-tournaments as a further generalization of local tournaments. Since then, the properties of in-tournaments, especially the cycle and path structure, have been investigated by Tewes [9][10][11], Tewes and Volkmann [12,13] and Meierling and Volkmann [7]. For more information concerning different generalizations of tournaments, the reader may be referred to the survey article of Bang-Jensen and Gutin [2].…”

Section: Definition 11 Let D Be a Digraph If The Longest Path Thromentioning

confidence: 99%

Solution of a conjecture of Volkmann on longest paths through an arc in strongly connected in‐tournaments

Meierling¹

2008

Journal of Graph Theory

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An in-tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. Let m = 4 or m = 5 and let D be a strongly connected in-tournament of order n ≥ 2m − 2 such that each arc belongs to a directed path of order at least m. In 2000, Volkmann showed that if D contains an arc e such that the longest directed path through e consists of exactly m vertices, then e is the only arc of D with that property. In this article we shall see that this proposition is true for m ≥ 4, thereby validating a conjecture of Volkmann. Furthermore, we prove that if we ease the restrictions on the order of D to n ≥ 2m − 3, the in-tournament D in question has at most two such arcs. In doing so, we also give a characterization of the in-tournaments with exactly two such arcs.

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“…Some problems concerning in-tournaments have been studied by Bang-Jensen et al in their initial article [4]. Later on, Tewes and Volkmann [13,14], Volkmann [15], Tewes [11,12], Peters and Volkmann [10] and Meierling and Volkmann [9] focused on the cycle and path structure of this class of digraphs. For more information concerning different generalizations of tournaments, the reader may refer to the survey article or the comprehensive monograph on digraphs of Bang-Jensen and Gutin [2,3].…”

Section: Terminology and Introductionmentioning

confidence: 99%

Solution of a conjecture of Tewes and Volkmann regarding extendable cycles in in‐tournaments

Meierling

2009

Journal of Graph Theory

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A directed cycle C of a digraph D is extendable if there exists a directed cycle C in D that contains all vertices of C and an additional one. In 1989, Hendry defined a digraph D to be cycle extendable if it contains a directed cycle and every non-Hamiltonian directed cycle of D is extendable. Furthermore, D is fully cycle extendable if it is cycle extendable and every vertex of D belongs to a directed cycle of length three. In 2001, Tewes and Volkmann extended these definitions in considering only directed cycles whose length exceed a certain bound 3 ≤ k 8n−17 31 , then D is fully (n− 4 +1 3 )-extendable. In this article we shall see that if 3 ≤ ≤ 8n−17 31 , every vertex of D belongs to a directed cycle of length n− 4 +1 3 , which means that D is fully (n− 4 +1 3 )-extendable. This confirms a conjecture

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Pancyclic in-tournaments

Cited by 11 publications

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Vertex 6-pancyclic in-tournaments

Vertex 6-pancyclic in-tournaments

Solution of a conjecture of Volkmann on longest paths through an arc in strongly connected in‐tournaments

Solution of a conjecture of Tewes and Volkmann regarding extendable cycles in in‐tournaments

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