PACKING AND DECOMPOSITIONS IN TRANSITIVE TOURNAMENTS – PhD THESIS

In this thesis we shall deal with oriented graphs. The motivation for us is a result by Sali and Simonyi (see also a short proof by Gyárfás) where the existence of the decomposition of transitive tournaments on two isomorphic graphs is shown. In this thesis we start to study a problem of packing in transitive tournaments and we consider decompositions and partitions of transitive tournaments. The aim of this work is to transfer the known results on the packing graphs into transitive tournaments. The subject of this hearing is so inspired by the results of Sali, Simony and Gyárfás results of research describing the transitive tournaments in selected areas. We find various conditions for the size of oriented graphs, and meeting where we can find a packing them in the transitive tournament. We give some characterizations of graphs, which there are decompositions of transitive tournaments. We will also consider the relationship of selfcomplementary graph with oriented graphs, which are packable together with their copy. In first chapter we give the definitions of concepts, which we will use, and quote the results of earlier studies on simple graphs, which motivates examining these hearing problems in the transitive tournaments. Let us note that the thesis is the first paper concerning packaging, decompositions and partitions transitive tournaments, which treats of oriented graphs. The second chapter provides an overview of the investigation packing in the transitive tournament two and three copies of a oriented graph, and packing two different oriented graphs equivalent to the first results discovered for simple graphs. In the third section of this chapter we present somewhat surprising result, namely to prove that the hypothesis of Milner and Welsh is true also for packing in the transitive tournament. The third chapter starts with a generalization of Gyárfás proof on the almost selfcomplementary graphs. Then we will discuss it results concerning packaging of selfcomplementary subgraph into transitive tournament, and finally present partitions transitive tournament on the isomorphic parts. The fourth chapter is dedicated to the decompositions of transitive tournament into oriented graphs of size at most four. It turns out that the problem of decomposition even on small graphs do not is simple and requires the use of different methods, depending the structure of the graph. The end of summarize what we managed to show that and suggest further lines of research undertaken by us issues.