---
_id: '1911'
abstract:
- lang: eng
text: 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>Δ, where Δ is the maximal degree of the graph. It was proven by the first
author using equivariant topology that if q>Δ 2 then the topological Tverberg
theorem still works. It is conjectured that q>KΔ 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.
acknowledgement: Patrik Norén gratefully acknowledges support from the Wallenberg
foundation
author:
- first_name: Alexander
full_name: Engström, Alexander
last_name: Engström
- first_name: Patrik
full_name: Noren, Patrik
id: 46870C74-F248-11E8-B48F-1D18A9856A87
last_name: Noren
citation:
ama: Engström A, Noren P. Tverberg’s Theorem and Graph Coloring. *Discrete &
Computational Geometry*. 2014;51(1):207-220. doi:10.1007/s00454-013-9556-3
apa: Engström, A., & Noren, P. (2014). Tverberg’s Theorem and Graph Coloring.
*Discrete & Computational Geometry*. Springer. https://doi.org/10.1007/s00454-013-9556-3
chicago: Engström, Alexander, and Patrik Noren. “Tverberg’s Theorem and Graph Coloring.”
*Discrete & Computational Geometry*. Springer, 2014. https://doi.org/10.1007/s00454-013-9556-3.
ieee: A. Engström and P. Noren, “Tverberg’s Theorem and Graph Coloring,” *Discrete
& Computational Geometry*, vol. 51, no. 1. Springer, pp. 207–220, 2014.
ista: Engström A, Noren P. 2014. Tverberg’s Theorem and Graph Coloring. Discrete
& Computational Geometry. 51(1), 207–220.
mla: Engström, Alexander, and Patrik Noren. “Tverberg’s Theorem and Graph Coloring.”
*Discrete & Computational Geometry*, vol. 51, no. 1, Springer, 2014,
pp. 207–20, doi:10.1007/s00454-013-9556-3.
short: A. Engström, P. Noren, Discrete & Computational Geometry 51 (2014) 207–220.
date_created: 2018-12-11T11:54:40Z
date_published: 2014-01-01T00:00:00Z
date_updated: 2021-01-12T06:54:01Z
day: '01'
department:
- _id: CaUh
doi: 10.1007/s00454-013-9556-3
intvolume: ' 51'
issue: '1'
language:
- iso: eng
month: '01'
oa_version: None
page: 207 - 220
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '5183'
scopus_import: 1
status: public
title: Tverberg's Theorem and Graph Coloring
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 51
year: '2014'
...