Compressed polytopes, initial ideals and complete multipartite graphs

Abstract

Convex polytopes arising from complete multipartite graphs and their toric ideals will be studied. First, it is proved that such toric ideals possess squarefree quadratic initial ideals. Second, we show that these convex polytopes are compressed and compute their $f$-vectors, Ehrhart polynomials and normalized volumes explicitly. Finally, all complete multipartite graphs which yield initial ideals coming from finite partially ordered sets will be classified.