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Let $G$ be a finite graph on the vertex set $[d] = \{1, ..., d \}$ with theedges $e_1, ..., e_n$ and $K[\tb] = K[t_1, ..., t_d]$ 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 $\tb^e = t_it_j$ such that $e =\{i, j\}$ is an edge of $G$. Let $K[\xb] = K[x_1, ..., x_n]$ be the polynomialring in $n$ variables over $K$ and define the surjective homomorphism $\pi :K[\xb] \to K[G]$ by setting $\pi(x_i) = \tb^{e_i}$ for $i = 1, ..., n$. Thetoric ideal $I_G$ of $G$ is the kernel of $\pi$. It will be proved that, givenintegers $f$ and $d$ with $6 \leq f \leq d$, there exist a finite connectednonbipartite graph $G$ on $[d]$ together with a reverse lexicographic order$<_{\rev}$ on $K[\xb]$ and a lexicographic order $<_{\lex}$ on $K[\xb]$ suchthat (i) $K[G]$ is normal, (ii) $\depth K[\xb]/\ini_{<_{\rev}}(I_G) = f$ and(iii) $K[\xb]/\ini_{<_{\lex}}(I_G)$ is Cohen--Macaulay, where$\ini_{<_{\rev}}(I_G)$ (resp.\ $\ini_{<_{\lex}}(I_G)$) is the initial ideal of$I_G$ with respect to $<_{\rev}$ (resp.\ $<_{\lex}$) and where $\depthK[\xb]/\ini_{<_{\rev}}(I_G)$ is the depth of $K[\xb]/\ini_{<_{\rev}}(I_G)$.

Depth of initial ideals of normal edge rings