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arXiv · 1209.4879

Coloring d-Embeddable k-Uniform Hypergraphs

Carl Georg Heise·Konstantinos Panagiotou·Oleg Pikhurko·Anusch Taraz
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Abstract

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.

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Carl Georg Heise, Konstantinos Panagiotou, Oleg Pikhurko, Anusch Taraz. 2014-11-27. Coloring d-Embeddable k-Uniform Hypergraphs. https://doi.org/10.1007/s00454-014-9641-2

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