Local Tournaments and In - Tournaments

Preface Tournaments constitute perhaps the most well-studied class of directed graphs. One of the reasons for the interest in the theory of tournaments is the monograph Topics on Tournaments [58] by Moon published in 1968, covering all results on tournaments known up to this time. In particular, three results deserve special mention: in 1934 Rédei [60] proved that every tournament has a directed Hamilto-nian path, in 1959 Camion [27] showed that every strongly connected tournament has a directed Hamiltonian cycle and in 1966 Moon [57] published his famous theorem which says that every strongly connected tournament on n vertices is vertex pancyclic, i.e. every vertex is contained in a directed cycle of length 3, 4,. .. , n. The latter result is a strong improvement of Camion's theorem and can be considered a milestone in the investigation of the cycle structure of tournaments. In the following years survey articles dealing with tournaments were contributed by All these articles kept the interest of mathematicians working in this field growing. There are various generalizations of tournaments: local tournaments and locally semicomplete digraphs, in-tournaments and locally in-semicomplete digraphs, mul-tipartite tournaments and semicomplete multipartite digraphs, quasi-transitive di-graphs and several others. For a comprehensive overview of multipartite tournaments and semicomplete multipartite digraphs the reader may be referred to the survey articles of Gutin [40] and Volkmann [75, 78], and for a summary of the remaining classes to the article of Bang-Jensen and Gutin [11]. It is a natural question to ask which of the results for tournaments can be extended to one of its generalized classes. In view of Rédei's and Moon's theorems, the path and cycle structure are specifically of interest. In this thesis, two different generalizations of tournaments are considered, namely the class of local tournaments and the class of in-tournaments. For the sake of simplicity, directed paths and directed cycles are called paths and cycles throughout this thesis. v vi Preface A digraph without loops, multiple arcs and cycles of length two is called a local tournament if the set of in-neighbors as well as the set of out-neighbors of every vertex induces a tournament. This class, or more generally the class of locally semicomplete digraphs whose members might have cycles of length two, was introduced by Bang-Jensen [3] in 1990. Three years later, Bang-Jensen, Huang and Prisner [16] introduced a further generalization of local tournaments, the class of in-tournaments, in claiming …