98 research outputs found
Coloring d-Embeddable k-Uniform Hypergraphs
This paper extends the scenario of the Four Color Theorem in the following
way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly)
embedded into R^d. We investigate lower and upper bounds on the maximum (weak
and strong) chromatic number of hypergraphs in H(d,k). For example, we can
prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak
chromatic number is Omega(log n/log log n), whereas the weak chromatic number
for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.Comment: 18 page
Transversals and colorings of simplicial spheres
Motivated from the surrounding property of a point set in
introduced by Holmsen, Pach and Tverberg, we consider the transversal number
and chromatic number of a simplicial sphere. As an attempt to give a lower
bound for the maximum transversal ratio of simplicial -spheres, we provide
two infinite constructions. The first construction gives infintely many
-dimensional simplicial polytopes with the transversal ratio exactly
for every . In the case of , this meets the
previously well-known upper bound tightly. The second gives infinitely
many simplicial 3-spheres with the transversal ratio greater than . This
was unexpected from what was previously known about the surrounding property.
Moreover, we show that, for , the facet hypergraph
of a -dimensional simplicial sphere
has the chromatic number , where is the number of vertices of . This
slightly improves the upper bound previously obtained by Heise, Panagiotou,
Pikhurko, and Taraz.Comment: 22 pages, 2 figure
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