Regular Tournaments Research Articles

Links between the Slater Index and the Ryser Index of Tournaments

Given a tournament T, the Slater index i(T) of T is the minimum number of arcs that must be reversed to make T transitive. The Ryser index τ˜(T) of T, defined from the out-degrees of T, measures a remoteness between T and the transitive tournaments of same order. In this paper, we study some links between i(T) and τ˜(T). More precisely, calling I(n, τ) the maximum value of i(T) over the set of tournaments on n vertices and such that τ˜(T)=τ, we compute an upper bound of I(n,τ) for every value of τ. Then we use this upper bound to study a conjecture stated by J.-C. Bermond on the regularity (i.e., the fact that all the out-degrees are equal or almost equal) of the tournaments with a maximum Slater index by showing that the out-degrees of such tournaments cannot be “too far” from the ones of the regular tournaments.

Trees with three leaves are ( n+1)-unavoidable

We prove that every tree of order n⩾5 with three leaves is ( n+1)-unavoidable. More precisely, we prove that every tree A with three leaves of order n is contained in every tournament T of order n+1 except if ( T; A) is ( R 5; S 3 +) or its dual, where R 5 is the regular tournament on five vertices and S 3 + is the outstar of degree three, i.e. the tree consisting of a root dominating three leaves. We then deduce that Sumner's conjecture is true for trees with four leaves, i.e. every tree of order n with four leaves is (2 n−2)-unavoidable.

Cycles Containing a Given Arc in Regular Multipartite Tournaments

In this paper we prove that if T is a regular n-partite tournament with n≥6, then each arc of T lies on a k-cycle for k=4,5,...,n. Our result generalizes theorems due to Alspach[1] and Guo[3] respectively.

The cycle structure of regular multipartite tournaments

A multipartite tournament is an orientation of a complete multipartite graph. A tournament is a multipartite tournament, each partite set of which contains exactly one vertex. Alspach (Canad. Math. Bull. 10 (1967) 283) proved that every regular tournament is arc-pancyclic. Although all partite sets of a regular multipartite tournament have the same cardinality, Alspach's theorem is not valid for regular multipartite tournaments. In this paper, we prove that if the cardinality common to all partite sets of a regular n-partite ( n⩾3) tournament T is odd, then every arc of T is in a cycle that contains vertices from exactly m partite sets for all m∈{3,4,…, n}. This result extends Alspach's theorem for regular tournaments to regular multipartite tournaments. We also examine the structure of cycles through arcs in regular multipartite tournaments.

Embedding Digraphs on Orientable Surfaces

We consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of compatible Euler tours or orthogonal A-trails by Andersen and by Bouchet. This prior work has mostly been limited to embeddings of Eulerian digraphs on predetermined surfaces and to digraphs with underlying graphs of maximum degree at most 4. In this paper, a foundation is laid for the study of all Eulerian digraph embeddings. Results are proved which are analogous to those fundamental to the theory of undirected graph embeddings, such as Duke's theorem [5], and an infinite family of digraphs which demonstrates that the genus range for an embeddable digraph can be any nonnegative integer given. We show that it is possible to have genus range equal to one, with arbitrarily large minimum genus, unlike in the undirected case. The difference between the minimum genera of a digraph and its underlying graph is considered, as is the difference between the maximum genera. We say that a digraph is upper-embeddable if it can be embedded with two or three regions and prove that every regular tournament is upper-embeddable.

Domination graphs of regular tournaments

Let ℘ ∗(m,n) denote the set of all graphs that are the union of m even paths and n nontrivial odd paths, and ℘(m,n) denote the set of all graphs that are the union of m even paths and n odd paths. In this paper, we show that if G is the domination graph of a regular tournament then G∈℘(m,n) or G is an odd cycle. Also we give a necessary and sufficient condition for G∈℘ ∗(m,n) to be the domination graph of a regular tournament. Constructions used in this paper will provide insight into the structure of a large class of regular tournaments.

An upper bound on the Perron value of an almost regular tournament matrix

We provide an asymptotic upper bound on the Perron value of an almost regular tournament matrix, improving upon an existing bound due to Friedland. Our approach employs techniques from contrained optimization.

The Many Names of (7, 3, 1)

In the world of discrete mathematics, we encounter a bewildering variety of topics with no apparent connection between them. But appearances are deceptive. For example, combinatorics tells us about difference sets, block designs, and triple systems. Geometry leads us to finite projective planes and Latin squares. Graph theory introduces us to round–robin tournaments and map colorings, linear algebra gives us (0, 1)–matrices, and quadratic residues are among the many pearls of number theory. We meet the torus, that topological curiosity, while visiting the local doughnut shop or tubing down a river. Finally, in these fields we encounter such names as Euler, Fano, Fischer, Hadamard, Heawood, Kirkman, Singer and Steiner. This is a story about a single object that connects all of these. Commonly known as (7, 3, 1), it is all at once a difference set, a block design, a Steiner triple system, a finite projective plane, a complete set of orthogonal Latin squares, a doubly regular round-robin tournament, a skew-Hadamard matrix, and a graph consisting of seven mutually adjacent hexagons drawn on the torus. We are going to investigate these connections. Along the way, we’ll learn about all of these topics and just how they are tied together in one object—namely, (7, 3, 1). We’ll learn about what all of those people have to do with it. We’ll get to know this object quite well! So let’s find out about the many names of (7, 3, 1).

Trees with three leaves are (n + l)-unavoidable

Abstract Abstract We prove that every oriented tree of order n ≥ 5 with three leaves is (n + 1)-unavoidable. More precisely, we prove that every tree A of order n with three leaves is contained in every tournament T of order n + 1 except if T is the regular tournament on five vertices and A the outstar (instar) of degree three, that is, the tree consisting of a root dominating (dominated by) three leaves.

On the Maximum Number of Cyclic Triples in Oriented Graphs

The maximum number of cyclic triples in an oriented graph of a given order is well known, being realized by the regular and near-regular tournaments. We consider the corresponding problem of determining the maximum number of cyclic triples in an oriented graph with a given number of arcs. A formula is found provided that the number of vertices is not too small relative to the number of arcs.

Oriented graphs generated by random points on a circle

Extending the cascade model for food webs, we introduce a cyclic cascade model which is a random generation model of cyclic dominance relations. Put n species as n points Q 1,Q 2,…, Q n on a circle. If the counterclockwise way from Q i to Q j on the circle is shorter than the clockwise way, we say Q i dominates Q j . Consider a tournament whose dominance relations are generated from the points on a circle by this rule. We show that when we take n mutually independently distributed points on the circle, the probability of getting a regular tournament of order 2r+1 as the largest regular tournament is equal to (n/(2r+1))/2 n-1. This probability distribution is for the number of existing species after a sufficiently long period, assuming a Lotka-Volterra cyclic cascade model.

Oriented graphs generated by random points on a circle

Extending the cascade model for food webs, we introduce a cyclic cascade model which is a random generation model of cyclic dominance relations. Put n species as n points Q1,Q2,…, Qn on a circle. If the counterclockwise way from Qi to Qj on the circle is shorter than the clockwise way, we say Qi dominates Qj. Consider a tournament whose dominance relations are generated from the points on a circle by this rule. We show that when we take n mutually independently distributed points on the circle, the probability of getting a regular tournament of order 2r+1 as the largest regular tournament is equal to (n/(2r+1))/2n-1. This probability distribution is for the number of existing species after a sufficiently long period, assuming a Lotka-Volterra cyclic cascade model.

The cycle structure of regular multipartite tournaments

The cycle structure of regular multipartite tournaments

Oriented Hamiltonian Paths in Tournaments: A Proof of Rosenfeld's Conjecture

We prove that with three exceptions, every tournament of order n contains each oriented path of order n. The exceptions are the antidirected paths in the 3-cycle, in the regular tournament on 5 vertices, and in the Paley tournament on 7 vertices.

On almost regular tournament matrices

Spectral and determinantal properties of a special class M n of 2n×2n almost regular tournament matrices are studied. In particular, the maximum Perron value of the matrices in this class is determined and shown to be achieved by the Brualdi–Li matrix, which has been conjectured to have the largest Perron value among all tournament matrices of even order. We also establish some determinantal inequalities for matrices in M n and describe the structure of their associated walk spaces.

Pancyclic out-arcs of a vertex in tournaments

Thomassen (J. Combin. Theory Ser. B 28, 1980, 142–163) proved that every strong tournament contains a vertex x such that each arc going out from x is contained in a Hamiltonian cycle. In this paper, we extend the result of Thomassen and prove that a strong tournament contains a vertex x such that every arc going out from x is pancyclic, and our proof yields a polynomial algorithm to find such a vertex. Furthermore, as another consequence of our main theorem, we get a result of Alspach (Canad. Math. Bull. 10, 1967, 283–286) that states that every arc of a regular tournament is pancyclic.

Outpaths in semicomplete multipartite digraphs

An outpath of a vertex x (an arc xy, respectively) in a digraph is a directed path starting at x ( xy, respectively) such that x dominates the endvertex of the path only if the endvertex also dominates x. Firstly, we show that if D is a strongly connected semicomplete n-partite ( n⩾3) digraph, then every vertex v of D has an outpath of length k−1 for all k∈{3,4,…, n}. Our result generalizes a theorem of Moon (Canad. Math. Bull. 9 (1966) 297–301) for tournaments. Secondly, we show that if T is a regular n-partite ( n⩾3) tournament, then every arc of T has an outpath of length k−1 for all k satisfying 3⩽ k⩽ n. This result extends a theorem of Alspach (Canad. Math. Bull. 10 (1967) 283–286) for regular tournaments to regular multipartite tournaments.

The cycle structure of regular multipartite tournaments

Abstract Extended Abstract A multipartite tournament is an orientation of a complete multipartite graph. A tournament is a multipartite tournament, each partite set of which contains exactly one vertex.

Asymptotic Enumeration of Eulerian Circuits in the Complete Graph

We determine the asymptotic behaviour of the number of Eulerian circuits in a complete graph of odd order. One corollary of our result is the following. If a maximum random walk, constrained to use each edge at most once, is taken on Kn, then the probability that all the edges are eventually used is asymptotic to e3/4n−½. Some similar results are obtained about Eulerian circuits and spanning trees in random regular tournaments. We also give exact values for up to 21 nodes.

A note on perron vectors for almost regular tournament matrices

Let T be an almost regular tournament matrix of order n with right perron vector w. We show that if the ith row sum of T is (n − 2) 2 and the jth row sum is n 2 , then w i < w j . Thus in the round robin competition corresponding to T, the ranking schemes of Kendall and Wei and of Ramanujacharyula agree with the ranking generated by the row sums of T.