11,943 research outputs found
A relative bound for independence
We prove an upper bound for the independence number of a graph in terms of
the largest Laplacian eigenvalue, and of a certain induced subgraph. Our bound
is a refinement of a well-known Hoffman-type bound.Comment: 10 pages; preprint, comments are welcom
Tverberg's theorem and graph coloring
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.Comment: To appear in Discrete and Computational Geometry, 13 pages, 1 figure.
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