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res:
bibo_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>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.@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: Alexander
foaf_name: Engström, Alexander
foaf_surname: Engström
- foaf_Person:
foaf_givenName: Patrik
foaf_name: Noren, Patrik
foaf_surname: Noren
foaf_workInfoHomepage: http://www.librecat.org/personId=46870C74-F248-11E8-B48F-1D18A9856A87
bibo_doi: 10.1007/s00454-013-9556-3
bibo_issue: '1'
bibo_volume: 51
dct_date: 2014^xs_gYear
dct_language: eng
dct_publisher: Springer@
dct_title: Tverberg's Theorem and Graph Coloring@
...