Abstract
Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties, they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conjectures. In this article, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids—a combinatorial abstraction of neighborly polytopes—of small rank and corank. In particular, if we denote by OM(r, n) the set of all oriented matroids of rank r and n elements, we determine all uniform neighborly oriented matroids in OM(5, ≤12), OM(6, ≤9), OM(7, ≤11), and OM(9, ≤12) and all possible face lattices of neighborly oriented matroids in OM(6, 10) and OM(8, 11). Moreover, we classify all possible face lattices of uniform 2-neighborly oriented matroids in OM(7, 10) and OM(8, 11). Based on the enumeration, we construct many interesting examples and test open conjectures.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
