  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.
