Tverberg's theorem and graph coloring

Details Tverberg's theorem and graph coloring Tverberg's theorem and graph coloring

2011-05-07

arxiv.org/abs/1105.1455v2

Mathematics

Combinatorics

Updated version, 11 pages

Scientific paper

The topological Tverberg theorem have been generalized in several directions by setting extra restrictions on the Tverberg partitions. This was initiated by Vrecica and Zivaljevic who used the chessboard complexes studied by them together with Bjorner and Lovasz. They were motivated both by combinatorial applications of a colored Tverberg theorem and the enumeration of Tverberg partitions. Restricted Tverberg partitions defined from 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.

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