Monday, June 27, 2011
Freie Universität Berlin
Institut für Informatik
Lecture - 14:15
In typical extremal problem one wants to determine maximum cardinality of discrete structure with certain prescribed properties. Probably the earliest such result was obtain 100 years ago by Mantel who computed the maximum number of edges in a triangle free graph on n vertices. This was generalized by Turan for all complete graphs and became a starting point of Extremal Graph Theory. In this talk we survey several classical problems and results in this area and present some interesting applications of Extremal Graph Theory to other areas of mathematics. We also describe a recent surprising generalization of Turan's theorem which was motivated by question in Computational Complexity.
Colloquium - 16:00
We study sufficient conditions for Hamiltonian cycles in hypergraphs, and obtain both Tur\'an- and Dirac-type results.
While the Tur\'an-type result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Dirac-type result yields just a sufficient condition relying solely on the minimum vertex degree.
Joint work with Yury Person and Wilma Weps.