Abstract:

The topological Tverberg theorem states that for any prime power $q$ and continuous map from a $(d+1)(q-1)$-simplex to $\mathbb{R}^d$, there are $q$ disjoint faces $F_i$ of the simplex whose images intersect.

Usually there are many ways to choose the disjoint faces and still get intersecting images. One indication of this, is that one can forbid certain pairs of vertices of the simplex to end up in the same face $F_i$ and anyway manage to find disjoint $F_i$:s whose images intersect. If we start off with a graph $T$ with the same vertex set as the simplex, and the topological Tverberg theorem works with no two vertices adjacent in $T$ placed in the same face $F_i$, then $T$ is a Tverberg graph.

Some Tverberg graphs are known from research by Hell, Schöneborn, and Ziegler. I will show that if the maximal degree of a graph is $\delta$, and $\delta(\delta+1) < q$, then it is a Tverberg graph.