SOME STRUCTURAL PROPERTIES OF BIPARTITE TOURNAMENTS (DIGRAPHS, GRAPHS)
Abstract
In this thesis some properties of bipartite tournaments, which are orientations of complete bipartite graphs, are investigated. Bipartite tournaments can also be represented by means of matrices of zeros and ones or by bipartite graphs. The interconnection between these representations is stressed throughout this work. A bipartite tournament may also be considered as a model of the outcomes of a competition between two teams, where every player in the first team opposes each player of the second and there are no ties. The converse of a bipartite tournament is obtained by reversing the direction of all the arcs. Necessary and sufficient conditions are found for a bipartite tournament to be isomorphic to its converse. If the players in each team are ranked according to the number of their wins, the presence of cycles indicates upsets. Explicit formulas for certain upsets are obtained in terms of scores, and bounds on their numbers are given. The numbers of various types of subtournaments of a bipartitie tournament are studied and sharp bounds are given in some cases. In some others, the problem of determining whether the bounds are sharp is shown to be related to the Hadamard conjecture. Several dominance properties of bipartite tournaments are also studied. In a round robin tournament, the existence of a vertex such that every other vertex has distance at most two from it and several of its properties are known. In fact, if the tournament has no dominant vertex then there are at least three vertices as above. Some analogous results for bipartite tournaments are established. It is known that there exist nonisomorphic bipartite tournaments with the same scores on their vertices. Those score lists are obtained which determine a bipartite tournament uniquely up to isomorphism. Many of the results obtained for bipartite tournaments are compared with the corresponding known results of ordinary tournaments and the similarities or the differences are noted.
Degree
Ph.D.
Subject Area
Mathematics
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