2025/07/08 by Shuai Huang, Huang, Shuai, Miller, Jasper +4
Computer Science · Mathematics · #05E45 #52B12 #52B35 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2507.06120
openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a relatively simple combinatorial characterization of simplicial d-spheres on d+4 vertices. Our criteria are given in terms of the intersection patterns of a simplicial complex's family of minimal non-faces. Namely, let Σ be a simplicial complex on d+4 vertices and let F be its family of minimal non-faces. Then Σ is a d-sphere if and only if |F|=n≥ 3 is odd and there is an ordering A0,…, An-1 of the minimal non-faces, indices taken modulo n, such that successive Ai are disjoint and the alternating ((n-1))/(2)-fold intersections Ai∩ Ai+2 ∩ Ai+4 ∩ ⋯ ∩ Ai+n-3 partition the vertex set.