Properties of K-tournaments

In this thesis, we investigate several properties of k-tournaments, where k 2 3. These properties fall into three broad areas. The first contains properties related to the ranking of the participants in a k-tournament, including a representation theorem for posets. The second contains properties related to the representation of a finite group as the automorphism group of a k-tournament, with varying restrictions on the desired representation. The third area answers questions about regularity in k-tournaments. Chapter 1 contains an introduction, and the definitions and notation. In Chapter 2, we consider the ranking of the participants in a k-tournament. We introduce the notions of transitivity and quasitransitivity in a k-tournament, each of which extends the notion of transitivity in a tournament in a natural way, and we prove that every k-tournament on a sufficiently large number of vertices contains a quasitransitive sub-k-tournament on a given number of vertices, thus extending the analogous result for tournaments. We then consider ranking the participants in a general k-tournament. We define, for a general k-tournament, a binary relation on its vertex set, which corresponds to a partial ranking of the participants. We then show that any finite poset with cardinality at least k + 1 can be represented by a k-tournament, in the sense that there is a k-tournament whose ranking relation is isomorphic to the given poset. The construction of this k-tournament suggests an interesting generalisation of the dimension of a poset. In Chapter 3, we investigate the automorphism group of a k-tournament. We begin by characterising those finite groups G for which there exists a k-tournament whose automorphism group is isomorphic to G. This extends the theorem of Moon (1964) which characterises the finite groups admitting a representation as the automorphism group of a tournament. We then consider the problem of finding the 'smallest' ktournament whose automorphism group is isomorphic to G, where we determine how 'small' a k-tournament is by the number of orbits of its automorphism group acting on its vertex set. With this definition of size, our goal is to characterise those finite groups admitting a regular representation as the automorphism group of a k-tournament. We first construct, for each admissible group G of order at least k, a k-tournament whose automorphism group is isomorphic to G and has two orbits of vertices. We then show that every admissible cyclic group of order at least k, and every admissible group which has a minimal generating set with at least k elements, admits a regular representation as the automorphism group of a k-tournament. Finally, in Chapter 4 we investigate regular and almost regular k-tournaments. We show that there is a regular k-tournament on n vertices if and only if n 2 k and (;) I 0 (mod n), and that there is an almost regular k-tournament on n vertices for all n and k satisfying n > k. We then provide some explicit constructions of regular and almost regular k-tournaments.