Hamiltonian Tournaments and Gorenstein Rings

Let Gnbe 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 eqdenote the q th unit coordinate vector of Rn. WriteP(Gn;ω) ⊂ Rn for the convex hull of the ( n 2) pointsei − ej, wherei → j is the direction of the edge { i, j } in the orientationω . It will be proved that, for n ≥ 5, the Ehrhart ring of the convex polytopeP(Gn;ω) is Gorenstein if and only if (Gn; ω) possesses a Hamiltonian cycle, i.e., a directed cycle of lengthn .