Abstract: In this talk, we mainly discuss three geometric minmax problems as
Colloquium - 16 Uhr s.t.
Abstract: Motivated by the Colin de Verdiere invariant \mu(G) we discuss two classes of graphs. The first class of graphs is defined as follows. If G is a graph, let C_2(G) denote the complex obtained from G by attaching to each circuit of G a disc. We say that a graph is 4-flat if C_2(G) can be embedded in 4-space. The suspension of a flat graph is a graph which is 4-flat. The graphs that are obtained from K_7 and K_{3,3,1,1} by Y\Delta- and \Delta Y-transformations are example of graphs which are not 4-flat. We conjecture that the 4-flat graphs are exactly those graphs with \mu(G) \leq 5.
The second class is formed by those graphs which can be embedded in 3-space such that no circuit is a nontrivial knot. Since this class is closed under taking minors, one problem is to find graphs for which such an embedding is not possible. Can we find a certificate which shows that such an embedding is not possible (but not necessary the other way around)? Generalizing work of Conway and Gordan, who showed how to find a certificate for certain graphs in the case when one deals with the Arf invariant, we show how to find a certificate in the case of a Vassiliev invariant. (Some parts are joint work with R. Pendavingh.)