Tverberg's theorem and graph coloring

TL;DR

This paper explores the interplay between topological Tverberg theorems and graph colorings, proving a fixed-parameter version of a conjecture relating the number of colors to graph degree for topological partitions.

Contribution

It establishes a fixed-parameter version of the conjecture that q>KΔ suffices for the topological Tverberg theorem, using shellability for connectivity proofs.

Findings

Proves a fixed-parameter version of the conjecture relating colors and maximum degree.

Uses shellability to strengthen topological connectivity results.

Connects graph coloring concepts with topological combinatorics.

Abstract

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in the same part. If the restrictions are too harsh, then the topological Tverberg theorem fails. The colored Tverberg theorem corresponds to graphs constructed as disjoint unions of small complete graphs. Hell studied the case of paths and cycles. In graph theory these partitions are usually viewed as graph colorings. As explored by Aharoni, Haxell, Meshulam and others there are fundamental connections between several notions of graph colorings and topological combinatorics. For ordinary graph colorings it is enough to require that the number of colors q satisfy q>Delta, where Delta is the maximal degree of the graph. It was proven by the first author using equivariant topology that if q>\Delta^2 then the topological Tverberg theorem still works. It is conjectured that q>K\Delta is also enough for some constant K, and in this paper we prove a fixed-parameter version of that conjecture. The required topological connectivity results are proven with shellability, which also strengthens some previous partial results where the topological connectivity was proven with the nerve lemma.