Proof Without loss of generality, we may assume $d'=d$ . It follows from [Reference Lin and Shen3, Lemma 5.1] and Proposition 2 that $\omega (D)<n-1$ for any maximal clique D of $\mathcal {G}(d,{\boldsymbol \alpha })$ . Therefore, the equivalence of (a) and (b) follows immediately from the discussion preceding [Reference Lin and Shen3, Lemma 5.1]. Consequently, it remains to prove (b).

There exists some ${\boldsymbol \beta } = (\beta _1, \dots , \beta _n) \in {\mathbb N}^n$ with $|{\boldsymbol \beta }| = d' - \left \lceil \frac {n}{2} \right \rceil $ such that $\widetilde {{\boldsymbol \alpha }} := {\boldsymbol \alpha } - {\boldsymbol \beta } = (\widetilde {\alpha }_1, \dots , \widetilde {\alpha }_n)$ preserves the nondecreasing order $1 \le \widetilde {\alpha }_1 \le \cdots \le \widetilde {\alpha }_n$ . To verify this, it suffices to note that $d' \le |{\boldsymbol \alpha }| - n$ and hence

$$\begin{align*}|\widetilde{{\boldsymbol \alpha}}| = |{\boldsymbol \alpha}| - \left(d' - \left\lceil \frac{n}{2} \right\rceil\right) = |{\boldsymbol \alpha}| - d' + \left\lceil \frac{n}{2} \right\rceil \ge n + \left\lceil \frac{n}{2} \right\rceil \ge n. \end{align*}$$

Given this ${\boldsymbol \beta }$ , set $\widetilde {d} := d' - |{\boldsymbol \beta }| = \left \lceil \frac {n}{2} \right \rceil $ . One easily checks that $\widetilde {d} \le |\widetilde {{\boldsymbol \alpha }}| - \widetilde {d}$ and $\widetilde {d} < n$ . Thus, by Proposition 3, there exists a maximal clique $\widetilde {C}$ of $\mathcal {G}(\widetilde {d}, \widetilde {{\boldsymbol \alpha }})$ satisfying $\omega (\widetilde {C}) = \left \lfloor n - \frac {n}{\widetilde {d}} \right \rfloor = n-2$ .

The translation map $\widetilde {{\boldsymbol a}} \mapsto \widetilde {{\boldsymbol a}}^\uparrow := \widetilde {{\boldsymbol a}} + {\boldsymbol \beta }$ from $\mathcal {G}(\widetilde {d}, \widetilde {{\boldsymbol \alpha }})$ to $\mathcal {G}(d, {\boldsymbol \alpha })$ is well-defined and injective. Moreover, this map preserves maximal cliques: if $\widetilde {A} = (\widetilde {{\boldsymbol a}}^1, \dots , \widetilde {{\boldsymbol a}}^n)$ is a maximal clique of $\mathcal {G}(\widetilde {d}, \widetilde {{\boldsymbol \alpha }})$ , then its image $\widetilde {A}^\uparrow := (\widetilde {{\boldsymbol a}}^1 + {\boldsymbol \beta }, \dots , \widetilde {{\boldsymbol a}}^n + {\boldsymbol \beta })$ is also a maximal clique of $\mathcal {G}(d, {\boldsymbol \alpha })$ .

Consider the equivalence classes $\mathcal {E}_{\widetilde {C}}$ in $\mathcal {G}(\widetilde {d},\widetilde {{\boldsymbol \alpha }})$ and $\mathcal {E}_{\widetilde {C}^{\uparrow }}$ in $\mathcal {G}(d,{\boldsymbol \alpha })$ , respectively. The invariants $\kappa _1, \kappa _2$ introduced in [Reference Lin and Shen3, Definition 4.12] coincide for these two equivalence classes, as verified below:

1. (i) First, the chain of inequalities $1 = \alpha _1 \ge \widetilde {\alpha }_1 \ge 1$ forces $\widetilde {\alpha }_1 = \alpha _1 = 1$ and consequently $\beta _1 = 0$ . Write the maximal clique $\widetilde {C}$ as $(\widetilde {{\boldsymbol c}}^1, \dots , \widetilde {{\boldsymbol c}}^n)$ . By the explicit construction in the proof of [Reference Lin and Shen3, Proposition 5.5], we may choose $\widetilde {C}$ such that $\widetilde {c}_1^1 = \widetilde {c}_2^1 = 1$ . Then for its image $\widetilde {C}^\uparrow = (\widetilde {{\boldsymbol c}}^1 + {\boldsymbol \beta }, \dots , \widetilde {{\boldsymbol c}}^n + {\boldsymbol \beta })$ , we have $\widetilde {c}_1^1 + \beta _1 = 1$ and $\widetilde {c}_2^1 + \beta _2 \ge 1$ . This shows that the invariant $\kappa _1$ is preserved under the translation map and equals $1$ for both equivalence classes.

2. (ii) Second, since ${\boldsymbol \alpha } = \widetilde {{\boldsymbol \alpha }} + {\boldsymbol \beta }$ , we have the following equivalences: $\widetilde {c}_n^1 = \widetilde {\alpha }_n - 1$ if and only if $\widetilde {c}_n^1 + \beta _n = \alpha _n - 1$ , and for each $i \in [n-1]$ , $\widetilde {c}_i^1 = \widetilde {\alpha }_i$ if and only if $\widetilde {c}_i^1 + \beta _i = \alpha _i$ . This shows that the invariant $\kappa _2$ is preserved under the translation map and coincides for both equivalence classes.

Furthermore, by [Reference Lin and Shen3, Definition 4.8], the translation map preserves the order compatibility of obstructions: if $i \triangleleft j$ in the poset of obstructions associated with $\mathcal {E}_{\widetilde {C}}$ , then either $i \triangleleft j$ or i and j are incompatible in the poset associated with $\mathcal {E}_{\widetilde {C}^\uparrow }$ .

Crucially, weaker constraints on the poset of obstructions afford greater flexibility in choosing total orders. Thus, we may endow $\mathrm {MC}(\mathcal {G}(d, {\boldsymbol \alpha }))$ and $\mathrm {MC}(\mathcal {G}(\widetilde {d}, \widetilde {{\boldsymbol \alpha }}))$ with compatible total orders such that the translation map is order-preserving: if $\widetilde {A} \prec \widetilde {C}$ are maximal cliques of $\mathcal {G}(\widetilde {d}, \widetilde {{\boldsymbol \alpha }})$ , then their images satisfy $\widetilde {A}^\uparrow \prec \widetilde {C}^\uparrow $ in $\mathcal {G}(d, {\boldsymbol \alpha })$ . Moreover, the translation map preserves the cardinality of set differences: $\#(\widetilde {A}\left\backslash\right.\widetilde {C}) = \#(\widetilde {A}^\uparrow\left\backslash\right.\widetilde {C}^\uparrow )$ for every maximal clique $\widetilde {A}$ of $\mathcal {G}(\widetilde {d}, \widetilde {{\boldsymbol \alpha }})$ . Combining these facts, we deduce that $\omega (\widetilde {C}^\uparrow ) \ge \omega (\widetilde {C}) = n-2$ . As we have already shown that $\omega (\widetilde {C}^\uparrow ) < n-1$ , we conclude that $\omega (\widetilde {C}^\uparrow ) = n-2$ , completing the proof.

Proof Let J denote the presentation ideal of $\mathcal {A}_{d,{\boldsymbol \alpha }}$ . Combining results from [Reference Lin and Shen3, Remark 2.5], [Reference Conca and Varbaro1, Corollary 2.7], and [Reference Herzog and Hibi2, Proposition 8.1.10], we have the chain of equalities:

$$\begin{align*}\mathrm{reg}(\mathcal{A}_{d,{\boldsymbol \alpha}}) = \mathrm{reg}({\mathbb K}[{\boldsymbol T}]/J) = \mathrm{reg}({\mathbb K}[{\boldsymbol T}]/\mathrm{in}(J)) = \mathrm{pd}((\mathrm{in}(J))^\vee). \end{align*}$$

The conclusion now follows directly from the corrected characterizations of projective dimension in Propositions 2–4.

Proof The algebra $\mathcal {A}_{d,{\boldsymbol \alpha }}$ is Cohen–Macaulay by [Reference Lin and Shen3, Remark 2.5]. Thus, the statements follow from Remark 2.4, Lemma 5.7, and Equation (8) of the original paper [Reference Lin and Shen3], as well as Theorem 5 above.