The Hilbert basis of the cut cone over the complete graph K 6

Abstract

Given a polyhedral cone C, generated by the integer vectors x 1,⋯, x n, the set of integer vectors of C is an additive semi-group, whose minimal set of generators (for linear cobinations with coefficients in ℤ+) is called the Hilbert basis of C. Integer points of C which are not in the integer cone ℤ+(x 1,⋯, x n) are called quasi-h points.The set of cuts on a graph forms a Hilbert basis for all graphs strictly smaller than K 6, but not for K 6.Two results are proven here: The Hilbert basis for the cut cone over K 6 is explicitly given: it consists of the 31 non-zero cuts and of the 15 vectors d e defined for each edge e by: d fe =2 for f ≠ e and d ee =4. The quasi-h points for K 6 are characterized: they are exactly the d e+nδ(υ) for υ non adjacent to e and n ∈ ℤ+. KeywordsComplete GraphToric VarietyIntersection VectorInteger PointInteger VectorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.