DEPTH OF INITIAL IDEALS OF NORMAL EDGE RINGS

Let G be a finite graph on the vertex set [d]={1,..., d} with the edges e(1),..., e(n) and K[t]=K[t(1),..., t(d)] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring K[G] which is generated by those monomials t (e)=t(i)t(j) such that e={i, j} is an edge of G. Let K[x]=K[x(1),..., x(n)] be the polynomial ring in n variables over K, and define the surjective homomorphism : K[x]K[G] by setting (x(i))=t (e)(i) for i=1,..., n. The toric ideal I-G of G is the kernel of . It will be proved that, given integers f and d with 6fd, there exists a finite connected nonbipartite graph G on [d] together with a reverse lexicographic order <(rev) on K[x] and a lexicographic order <(lex) on K[x] such that (i) K[G] is normal with Krull-dimK[G]=d, (ii) depthK[x]/in(<rev) (I-G)=f and K[x]/in(<lex) (I-G) is Cohen-Macaulay, where in(<rev) (I-G) (resp., in(<lex) (I-G)) is the initial ideal of I-G with respect to <(rev) (resp., <(lex)) and where depthK[x]/in(<rev) (I-G) is the depth of K[x]/in(<rev) (I-G).