Hamiltonian Tournaments and Gorenstein Rings

Let Gn be the complete graph on the vertex set [n] = {1, 2, . . . , n} and ω an orientation of Gn , i.e., ω is an assignment of a direction i → j of each edge {i, j} of Gn . Let eq denote the qth unit coordinate vector of Rn . Write P(Gn ;ω) ⊂ R n for the convex hull of the (n 2 ) points ei − e j , where i → j is the direction of the edge {i, j} in the orientation ω. It will be proved that, for n ≥ 5, the Ehrhart ring of the convex polytope P(Gn ;ω) is Gorenstein if and only if (Gn;ω) possesses a Hamiltonian cycle, i.e., a directed cycle of length n.