Gröbner bases and triangulations of the second hypersimplex

Abstract

The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex Δ(2, n). We present a quadratic Gröbner basis for the associated toric ideal K(Kn). The simplices in the resulting triangulation of Δ(2, n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding of Kn. For n≥6 the number of distinct initial ideals of I(Kn) exceeds the number of regular triangulations of Δ(2, n); more precisely, the secondary polytope of Δ(2, n) equals the state polytope of I(Kn) for n≤5 but not for n≥6. We also construct a non-regular triangulation of Δ(2, n) for n≥9. We determine an explicit universal Gröbner basis of I(Kn) for n≤8. Potential applications in combinatorial optimization and random generation of graphs are indicated. © 1995 Akadémiai Kiadó.