by assumption $G$ has no isolated vertices. So there exists $j'\in [n]\setminus V(C)$ such that $\{j, j'\} \notin E(\bar{G})$.

We may assume that $\{j', i\}\in E(\bar{G})$ for all $1\leq i\leq t+3$, because otherwise, as we have seen before for $j$, there exists $W\subset [n]$ with $|W|=t+2$ such that $(\bar{G})_W$ is not connected. Now let $W=V(C)\cup\{ j, j'\}$. As we mentioned in the proof of Lemma \[index\], $\widetilde{H}_2(\Delta_{W}; K)\neq 0$ and so $\beta_{t+1, t+5}\neq 0$.

\(b) Since $\index(I)=t$, [@EGHP Theorem 2.1] implies that there exists a minimal cycle of length $t+3$ in $\bar{G}$, say $C$. Let $\Delta=\Delta(\bar{G})$. Then $\widetilde{H}_1(\Delta_{V(C)}; K)\neq 0$. Hochster’s formula implies that $\beta_{t,t+3}(I)\geq1$. Since $n=t+3$, the only $W\subseteq [n]$ with $|W|=t+3$ is $V(C)$, and so $\beta_{t,t+3}(I)=1$, again by Hochster’s formula.

Now we are ready to prove the main theorem of this section.

The implication (a)(b) and also $\projdim(I)=n-3$ follows from [@EGHP Example 2.2].

(b)(a): Let $\index (I)=t$. By [@EGHP Theorem 2.1], $\bar{G}$ contains a minimal cycle of length $t+3$. Proposition \[beta\] implies that $G$ has $t+3$ vertices and so does $\bar{G}$. Hence in $\bar{G}$ there are no other vertices. Therefore $\bar{G}$ is a minimal cycle of length $t+3$. Moreover, $\projdim (I)=n-3$.

The following result supports our conjecture that 