Depth of initial ideals of normal edge rings

Let $G$ be a finite graph on the vertex set $[d]=\{1,\mathrm{.}\mathrm{.}\mathrm{.},d\}$ with the edges $e_1,\mathrm{.}\mathrm{.}\mathrm{.},e_n$ and $K[\text{tb}]=K[t_1,\mathrm{.}\mathrm{.}\mathrm{.},t_d]$ 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 ${\text{tb}}^e=t_i{t}_j$ such that $e = {i, j}$isanedgeof$G$\mathrm{.}Let$K[\xb] = K[x_1, …, x_n]$ be the polynomial ring in $n$ variables over $K$ and define the surjective homomorphism $\pi : K[\xb] \to K[G]$bysetting$\pi(x_i) = \tb^{e_i}$for$i = 1, …, n$. The toric ideal $I_G$ of $G$ is the kernel of $\pi$. It will be proved that, given integers $f$ and $d$ with $6\le f\le d$, there exist a finite connected nonbipartite graph $G$ on $[d]$ together with a reverse lexicographic order ${<}_{\text{rev}}$ on $K[\text{xb}]$ and a lexicographic order ${<}_{\text{lex}}$ on $K[\text{xb}]$ such that (i) $K[G]$ is normal, (ii) $\text{depth}K[\text{xb}]\mathrm{/}{\text{ini}}_{{<}_{\text{rev}}}(I_G)=f$ and (iii) $K[\text{xb}]\mathrm{/}{\text{ini}}_{{<}_{\text{lex}}}(I_G)$ is Cohen–Macaulay, where ${\text{ini}}_{{<}_{\text{rev}}}(I_G)$ (resp.\ ${\text{ini}}_{{<}_{\text{lex}}}(I_G)$) is the initial ideal of $I_G$ with respect to ${<}_{\text{rev}}$ (resp.\ ${<}_{\text{lex}}$) and where $\depth K[\xb]/\ini_{<{\rev}}(I_G)$isthedepthof$K[\xb]/\ini{<_{\rev}}(I_G)$.