Abstract
This paper extends the scenario of the Four Color Theorem in the following way. Let fancyscript{H}_{d,k} be the set of all k-uniform hypergraphs that can be (linearly) embedded into mathbb {R}^d. We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in fancyscript{H}_{d,k}. For example, we can prove that for dge 3 there are hypergraphs in fancyscript{H}_{2d-3,d} on n vertices whose chromatic number is Omega (log n/log log n), whereas the chromatic number for n-vertex hypergraphs in fancyscript{H}_{d,d} is bounded by {mathcal {O}}(n^{(d-2)/(d-1)}) for dge 3.
Highlights
The Four Color Theorem [1,2] asserts that every graph that is embeddable in the plane has chromatic number at most four
We focus our attention on hypergraphs, which are in general not embeddable into any specific dimension
We have decided to focus on linear embeddings, as they lead to a very accessible type of geometry and, at least in theory, the decision problem of whether a given k-uniform hypergraph is d-embeddable is decidable and in PSPACE [23]
Summary
The Four Color Theorem [1,2] asserts that every graph that is embeddable in the plane has chromatic number at most four. Grünbaum and Sarkaria (see [15,26]) have considered a different generalization of graph colorings to simplicial complexes by coloring faces They bound this face-chromatic number subject to embeddability constraints. We have decided to focus on linear embeddings, as they lead to a very accessible type of geometry and, at least in theory, the decision problem of whether a given k-uniform hypergraph is d-embeddable is decidable and in PSPACE [23]. We have demonstrated that for any 2 ≤ k ≤ d + 1 ≤ n there exists a k-uniform hypergraph on n vertices that is linearly d-embeddable and has strong chromatic number n.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
