2023/05/02 by Isabella Novik, Novik, Isabella, Hailun Zheng +1
Engineering · Mathematics · #52B05 #52B11 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2305.01829
openalex publication_date 2023/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define a certain merging operation that given two d-polytopes P and Q such that P has a simplex facet F and Q has a simple vertex v produces a new d-polytope P\hspace0.1em\triangleright Q with f0(P)+f0(Q)-(d+1) vertices. We show that if for some 1≤ i≤ d-1, P and Q are (d-i)-simplicial i-simple d-polytopes, then so is P\hspace0.1em\triangleright Q. We then use this operation to construct new families of (d-i)-simplicial i-simple d-polytopes. Specifically, we prove that for all 2≤ i ≤ d-2≤ 6 with the exception of (i,d)=(3,8) and (5,8), there is an infinite family of (d-i)-simplicial i-simple d-polytopes; furthermore, for all 2≤ i≤ 4, there is an infinite family of self-dual i-simplicial i-simple 2i-polytopes. Finally, we show that for any d≥ 4, there are 2Ω(N) combinatorial types of (d-2)-simplicial 2-simple d-polytopes with at most N vertices.