Polytopes with low excess degree

摘要: We study the existence and structure of -polytopes for which the number of edges is small compared to the number of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as . We show that the excess degree of a -polytope cannot lie in the range , complementing the known result that values in the range are impossible. In particular, many pairs are not realised by any polytope. For -polytopes with excess degree , strong structural results are known; we establish comparable results for excess degrees , , and . Frequently, in polytopes with low excess degree, say at most , the nonsimple vertices all have the same degree and they form either a face or a missing face. We show that excess degree is possible only for , or , complementing the known result that an excess degree is possible only for or .