Acyclic Subgraphs in $k$Majority Tournaments

2010

Few families of tournaments satisfying the n-e.c. adjacency property are known. We supply a new random construction for generating infinite families of vertex-transitive n-e.c. tournaments by considering circulant tournaments. Switching is used to generate new n-e.c. tournaments of certain orders. With aid of a computer search, we demonstrate that there is a unique minimum order 3-e.c. tournament of order 19, and there are no 3-e.c. tournaments of orders 20, 21, and 22. We show that there are no 4-e.c. tournaments of orders 47 and 48 improving the lower bound for the minimum order of such a tournament.

A graph or digraph is hamiltonian if it contains a cycle that visits every vertex, and traceable if it contains a path that visits every vertex. A (di)graph is k-traceable if each of its induced subdigraphs of order k is traceable. A digraph D is strong if for every pair u, v of vertices in D there is a directed path from u to v and a directed path from v to u.

Discrete Mathematics, 2019

A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erdős and J. W. Moon. By probabilistic methods, the existence of "unrankable" tournaments was proved. On the other hand, they also mentioned the problem of explicit constructions. However, there seems to be only a few of explicit constructions of such tournaments. In this note, we give a construction of many such tournaments by using skew Hadamard difference sets which have been investigated in combinatorial design theory. 2010 Mathematics Subject Classification. 05C20.

Discrete & Computational Geometry, 2013

We study a high-dimensional analog for the notion of an acyclic (aka transitive) tournament. We give upper and lower bounds on the number of d-dimensional n-vertex acyclic tournaments. In addition, we prove that every n-vertex d-dimensional tournament contains an acyclic subtournament of Ω(log 1/d n) vertices and the bound is tight. This statement for tournaments (i.e., the case d = 1) is a well-known fact. We indicate a connection between acyclic high-dimensional tournaments and Ramsey numbers of hypergraphs. We investigate as well the interrelations among various other notions of acyclicity in high-dimensional tournaments. These include combinatorial, geometric and topological concepts.

Journal of Graph Theory, 1998

We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly non-trivial, even for these "tournament-like" digraphs.

Order

In an earlier paper (see Sali and Simonyi Eur. J. Combin. 20, 93-99, 1999) the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositions of odd vertex complete graphs to Hamiltonian cycles admit such an orientation. These conditions are further generalized and some necessary conditions are given as well.

Journal of Algorithms, 1983

Given a tournament with n vertices, we consider the number of comparisons needed, in the worst case, to find a permutation o ,02... u,, of the vertices, such that the results of the games u,v2, vzuj,.. , on-,",, match a prescribed pattern. If the pattern requires all arcs to go forward, i.e., u,-+ 02, va + 03,.. , v,-i + v,,, and the tournament is transitive, then this is essentially the problem of sorting a linearly ordered set. It is well known that the number of comparisons required in this case is at least cnlg n, and we make the observation that O(nig n) comparisons suffice to find such a path in any (not necessarily transitive) tournament. On the other hand, the pattern requiring the arcs to alternate backward-fotward-backward, etc., admits an algorithm for which O(n) comparisons always suffice. Our main result is the somewhat surprising fact that for various other patterns the complexity (number of comparisons) of finding paths matching the pattern can be cn Ig"n for any a! between 0 and 1. Thus there is a veritable spectrum of complexities, depending on the prescribed pattern of the desired path. Similar problems on complexities of algorithms for finding Hamiltonian cycles in graphs and directed graphs have been considered by various authors, [2, pp. 142, 148, 149; 41. DEFINITIONS A tournament is an orientation of a complete graph. Any two vertices (players) u, w are adjacent by exactly one arc, i.e., either u-, w (u beats w), or w-B o (u loses to w); there are no ties. Throughout the paper T, denotes an arbitrary tournament with vertices 1,2,.. . , n. A tournament is transitive if the relation u-) w is transitive. Clearly, a transitive tournament is linearly ordered by the relation o + w, and there is (up to isomorphism) a unique transitive tournament on 1,2,.. . , n;ithasi-,jifandonlyifi>j,anditis denoted by TT,.

Journal of Combinatorial Theory, Series B, 1991

An n-partite tournament, n 3 2, or multipartite tournament is an oriented graph obtained by orienting each edge of a complete n-partite graph. The cycle structure of multipartite tournaments is investigated and properties of vertices with maximum score are studied.

Canadian mathematical bulletin, 1970

Journal of Graph Theory, 1999

It is well known that every tournament contains a Hamiltonian path, which can be restated as that every tournament contains a unary spanning tree. The purpose of this article is to study the general problem of whether a tournament contains a k-ary spanning tree. In particular, we prove that, for any fixed positive integer k, there exists a minimum number h(k) such that every tournament of order at least h(k) contains a k-ary spanning tree. The existence of a Hamiltonian path for any tournament is the same as h(1) = 1. We then show that h(2) = 4 and h(3) = 8. The values of h(k) remain unknown for k ≥ 4.

Algorithmica, 2013

The k-feedback arc set problem is to determine whether there is a set F of at most k arcs in a directed graph G such that the removal of F makes G acyclic. The k-feedback arc set problems in tournaments and bipartite tournaments (k-FAST and k-FASBT) have applications in ranking aggregation and have been extensively studied from the viewpoint of parameterized complexity. By introducing a new concept called "bimodule," we provide a problem kernel with O(k 2 ) vertices for k-FASBT, which improves the previous result by a factor of k.

Electronic Notes in Discrete Mathematics, 2016

Let k ≥ 2 be an integer. Bermond and Thomassen conjectured that every digraph with minimum out-degree at least 2k − 1 contains k vertex-disjoint cycles. Recently Bai, Li and Li proved this conjecture for bipartite digraphs. In this paper we prove that every bipartite tournament with minimum out-degree at least 2k − 2, minimum in-degree at least 1 and partite sets of cardinality at least 2k contains k vertex-disjoint 4-cycles whenever k ≥ 3. Finally, we show that every bipartite tournament with minimum degree δ = min{δ + , δ − } at least 1.5k − 1 contains at least k vertex-disjoint 4-cycles.

The Electronic Journal of Combinatorics, 1995

We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $n$ vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds $s(n) \geq 3n - 2 - \lfloor \log_2 n \rfloor$ or $s(n) \geq 3n - 1 - \lfloor \log_2 n \rfloor$, depending on the binary expansion of $n$. When $n = 2^k - 2^t$ we show that the bounds are tight with $s(n) = 3n - 2 - \lfloor \log_2 n \rfloor$. One view of this problem is that if the 'teams' in a tournament are ranked to minimize inconsistencies there is some tournament with $s(n)$ 'teams' in which $n$ are ranked wrong. We will also pose some questions about conditions on feedback arc sets, motivated by our proofs, which ensure equality between the maximum number of arc disjoint cycles and the minimum size of a feedback arc set in a tournament.

Discrete Mathematics, 1998

A multipartite tournament is an orientation of a complete multipartite graph. Simple derivations are obtained of the numbers of unlabeled acyclic and unicyclic multipartite tournaments, and unlabeled bipartite tournaments with exactly k cycles, which are pairwise vertex-disjoint.

Social Choice and Welfare

Tournament solutions provide methods for selecting the "best" alternatives from a tournament and have found applications in a wide range of areas. Previous work has shown that several well-known tournament solutions almost never rule out any alternative in large random tournaments. Nevertheless, all analytical results thus far have assumed a rigid probabilistic model, in which either a tournament is chosen uniformly at random, or there is a linear order of alternatives and the orientation of all edges in the tournament is chosen with the same probabilities according to the linear order. In this work, we consider a significantly more general model where the orientation of different edges can be chosen with different probabilities. We show that a number of common tournament solutions, including the top cycle and the uncovered set, are still unlikely to rule out any alternative under this model. This corresponds to natural graph-theoretic conditions such as irreducibility of the tournament. In addition, we provide tight asymptotic bounds on the boundary of the probability range for which the tournament solutions select all alternatives with high probability.