Monomial Rings of Maximal Cliques of Graphs and Tdi Systems

Let A be an integral matrix whose set of column vectors is A = {v1, . . . , vq} and let A(P ) be the Ehrhart ring of P = conv(A). We are able to show that if A is the incidence matrix of a d-uniform unmixed clutter with covering number g and the system x ≥ 0;xA ≥ 1 is TDI, then the Castelnuovo-Mumford regularity of A(P ) is sharply bounded by (d − 1)(g − 1). Let R = K[x1, . . . , xn] be a polynomial ring over a field K, where n is the number of rows of A. If A is the vertex-clique matrix of a Meyniel graph G, we prove the equality K[x1 t, . . . , xq t] = A(P ). As a consequence we show that x ≥ 0;xA ≥ 1 is a TDI system if and only if Ii = I, where I ⊂ R is the edge ideal of the clique-clutter of G, I is the ith symbolic power of I and Ii is the integral closure of I. Then we study TDI systems. If {x| xA ≤ w} is integral for some integral vector w = (wi) and H = {(v1, w1), . . . , (vq, wq)} is a Hilbert basis, we show that the system xA ≤ w is TDI. If A is non-negative, we characterize when the system x ≥ 0; xA ≤ w is TDI.