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 idealK(K n ). 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 ofK n . Forn≥6 the number of distinct initial ideals ofI(K n ) exceeds the number of regular triangulations of Δ(2,n); more precisely, the secondary polytope of Δ(2,n) equals the state polytope ofI(K n ) forn≤5 but not forn≥6. We also construct a non-regular triangulation of Δ(2,n) forn≥9. We determine an explicit universal Gröbner basis ofI(K n ) forn≤8. Potential applications in combinatorial optimization and random generation of graphs are indicated.

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Research partially supported by a doctoral fellowship of the National University of Mexico, the National Science Foundation, the David and Lucile Packard Foundation and the U.S. Army Research Office (through ACSyAM/MSI, Cornell).

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De Loera, J.A., Sturmfels, B. & Thomas, R.R. Gröbner bases and triangulations of the second hypersimplex. Combinatorica 15, 409–424 (1995). https://doi.org/10.1007/BF01299745

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Mathematics Subject Classification (1991)

  • 52 B 12
  • 13 P 10
  • 05 C 50