Depth of Initial Ideals of Normal Edge Rings

Let G be a finite graph on the vertex set [d] = {1, . . . , d} with the edges e1, . . . , en and K[t] = K[t1, . . . , td] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ringK[G] which is generated by those monomials t = titj such that e = {i, j} is an edge of G. Let K[x] = K[x1, . . . , xn] be the polynomial ring in n variables over K and define the surjective homomorphism π : K[x] → K[G] by setting π(xi) = t ei for i = 1, . . . , n. The toric ideal IG of G is the kernel of π. It will be proved that, given integers f and d with 6 ≤ f ≤ d, there exist 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, (ii) depthK[x]/ in<rev (IG) = f and (iii) K[x]/ in<lex(IG) is Cohen–Macaulay, where in<rev (IG) (resp. in<lex(IG)) is the initial ideal of IG with respect to <rev (resp. <lex) and where depthK[x]/ in<rev (IG) is the depth of K[x]/ in<rev (IG).