Arc-pancyclicity in multipartite tournaments & GTECS

This thesis consists of two parts: The first part deals with the theory of pancyclic arcs in multipartite tournaments and the second part contains a nice application of graph theory in crystallography. Tournaments are the most interesting class of digraphs. Particularly their cycle structure is an intensively studied topic. A famous result by Moon shows that every vertex in a strong tournament is pancyclic, i. e. it is contained in cycles of all possible lengths. The theoretical part of this thesis presents various results concerning the number of pancyclic arcs in tournaments. Furthermore, an overview for another interesting problem in tournaments called „the number of vertices whose all out-arcs are pancyclic“ is given and a first corresponding result to this problem in multipartite tournaments, a well-known superclass of tournaments, is shown. In nowadays' research of chemistry or crystallography, the interest in visualisation and analysis of large crystal structures grows continuously. Therefore, the second part of this thesis describes the graph theoretical ideas and algorithms using (infinite) periodic graphs behind GTECS (GraphTheoretical Evaluation of Crystal Structures), a new powerful tool to interpret complicated extended structures. It is the result of a joint „Seed Fund“-project carried out by chemists, computer scientists and mathematicians at the RWTH Aachen University.