2017/03/30 by Pineda-Villavicencio, Guillermo, Ugon, Julien, Yost, David · 1 citation
#52B05 (Primary) #52B11 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1703.10702
We define the excess degree ξ(P) of a d-polytope P as 2f1-df0, where f0 and f1 denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple. It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d-2, and the value d-1 only occurs when d=3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even. The excess degree is then applied in three different settings. It is used to show that polytopes with small excess (i.e. ξ(P)