Complete Disorder is Impossible: The Mathematical Work of Walter Deuber

This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Complete disorder is impossible – this theme of Ramsey Theory, as stated by Theodore S. Motzkin, was a guiding theme throughout Walter Deuber's scientific life.Peer Reviewe

Counting Partial Orders with a Fixed Number of Comparable Pairs

This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.In 1978, Dhar suggested a model of a lattice gas whose states are partial orders. In this context he raised the question of determining the number of partial orders with a fixed number of comparable pairs. Dhar conjectured that in order to find a good approximation to this number, it should suffice to enumerate families of layer posets. In this paper we prove this conjecture and thereby prepare the ground for a complete answer to the question.Peer Reviewe

On random planar graphs, the number of planar graphs and their triangulations

AbstractLet Pn be the set of labelled planar graphs with n vertices. Denise, Vasconcellos and Welsh proved that |Pn|⩽n!(75.8)n+o(n) and Bender, Gao and Wormald proved that |Pn|⩾n!(26.1)n+o(n). Gerke and McDiarmid proved that almost all graphs in Pn have at least 13/7n edges. In this paper, we show that |Pn|⩽n!(37.3)n+o(n) and that almost all graphs in Pn have at most 2.56n edges. The proof relies on a result of Tutte on the number of plane triangulations, the above result of Bender, Gao and Wormald and the following result, which we also prove in this paper: every labelled planar graph G with n vertices and m edges is contained in at least ε3(3n−m)/2 labelled triangulations on n vertices, where ε is an absolute constant. In other words, the number of triangulations of a planar graph is exponential in the number of edges which are needed to triangulate it. We also show that this bound on the number of triangulations is essentially best possible

Verbundprojekt PARALOR: Parallele Verfahren zur Wegoptimierung in Flugplanung und Logistik

Die Lösung kombinatorischer Optimierungsprobleme ist in vielen Bereichen von Wirtschaft und Technik der Schlüssel zur Steigerung der Effizienz technischer Abläufen, zur Verbesserung der Produktqualität und zur Veringerung von Produktions-, Material- und Transportkosten. Der Einsatz herkömmlicher sequentieller Verfahren ist für praxisrelevante Probleme aufgrund der enormen Rechenzeiterfordernisse nur sehr eingeschränkt möglich. Parallele Systeme bieten eine Möglichkeit, derartige Probleme in vertretbarer Zeit zu lösen. Im Rahmen des Verbundprojektes PARALOR wird untersucht, wie parallele Algorithmen der kombinatorischen Optimierung in konkreten, industriellen Anwendungen aus der Flugplanung sowie der Speditionslogistik effizient eingesetzt werden können. In diesem Artikel werden wesentliche Ergebnisse des Projekts exemplarisch vorgestellt