2021/11/12 by Joseph Briggs, Briggs, Joseph, Michael Gene Dobbins +3
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2111.06560
openalex publication_date 2021/11/12 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Motivated from the surrounding property of a point set in ℝd 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 d-spheres, we provide two infinite constructions. The first construction gives infintely many (d+1)-dimensional simplicial polytopes with the transversal ratio exactly (2)/(d+2) for every d≥ 2. In the case of d=2, this meets the previously well-known upper bound 1/2 tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than 1/2. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for d≥ 3, the facet hypergraph F(K) of a d-dimensional simplicial sphere K has the chromatic number χ(F(K)) ∈ O(n(\lceil d/2\rceil-1)/(d)), where n is the number of vertices of K. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz.