A local criterion for Tverberg graphs

TL;DR

This paper introduces a local criterion for identifying Tverberg graphs, expanding the class of such graphs by linking their maximal degree to the parameters of the topological Tverberg theorem.

Contribution

It establishes a new local condition based on maximal degree that guarantees a graph is Tverberg, broadening known classes of Tverberg graphs.

Findings

Maximal degree D satisfying D(D+1)<q implies the graph is Tverberg.

New examples of Tverberg graphs are identified using the local criterion.

Results apply to both topological and affine Tverberg theorems.

Abstract

The topological Tverberg theorem states that for any prime power q and continuous map from a (d+1)(q-1)-simplex to R}^d, there are q disjoint faces F_i of the simplex whose images intersect. It is possible to put conditions on which pairs of vertices of the simplex that are allowed to be in the same face F_i. A graph with the same vertex set as the simplex, and with two vertices adjacent if they should not be in the same F_i, is called a Tverberg graph if the topological Tverberg theorem still work. These graphs have been studied by Hell, Schoneborn and Ziegler, and it is known that disjoint unions of small paths, cycles, and complete graphs are Tverberg graphs. We find many new examples by establishing a local criterion for a graph to be Tverberg. An easily stated corollary of our main theorem is that if the maximal degree of a graph is D, and D(D+1)<q, then it is a Tverberg graph. We state the affine versions of our results and also describe how they can be used to enumerate Tverberg partitions.