Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems
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Given a complete simple topological graph G, a k-face generated by G is the open bounded region enclosed by the edges of a non-self-intersecting k-cycle in G. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete n-vertex simple topological graph generates at least Ω(n^1/3) pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on n vertices drawn in the unit square generates a 4-face with area at most O(n^-1/3). Finally, we investigate a ℤ_2 variant of Heilbronn triangle problem.
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