2021/04/30 by Riccardo W. Maffucci, Maffucci, Riccardo W.
Computer Science · Engineering · Mathematics · #05C07 #05C10 #05C75 #05C85 #52B05 #52B10 #Advanced Graph Theory Research #Combinatorics #Combinatorics (math.CO) #Degree (music) #Discrete mathematics #FOS: Mathematics #Graph #Limits and Structures in Graph Theory #Mathematics #Optimization and Packing Problems #Physics #Polytope #Vertex (graph theory) #Vertex connectivity #math.CO #msc:05C07 #msc:05C10 #msc:05C75 #msc:05C85 #msc:52B05 #msc:52B10
paper · pdf · doi:10.48550/arxiv.2105.00022
arxiv created 2021/04/30 · openalex publication_date 2021/04/30 · arxiv updated 2021/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let n≥ 3 and rn be a 3-polytopal graph such that for every 3≤ i≤ n, rn has at least one vertex of degree i. We find the minimal vertex count for rn. We then describe an algorithm to construct the graphs rn. A dual statement may be formulated for faces of 3-polytopes. The ideas behind the algorithm generalise readily to solve related problems. Moreover, given a 3-polytope tl comprising a vertex of degree i for all 3≤ i≤ l, l fixed, we define an algorithm to output for n>l a 3-polytope tn comprising a vertex of degree i, for all 3≤ i≤ n, and such that the initial tl is a subgraph of tn. The vertex count of tn is asymptotically optimal, in the sense that it matches the aforementioned minimal vertex count up to order of magnitude, as n gets large. In fact, we only lose a small quantity on the coefficient of the second highest term, and this quantity may be taken as small as we please, with the tradeoff of first constructing an accordingly large auxiliary graph.