= Solution
For each $v\in V(\Gamma)$, the <vertex link of a Salvetti complex> has two vertices $v^+$ and $v^-$. If $v$ has degree $d_\Gamma(v)$, then each of $v^+$ and $v^-$ is adjacent to both signed vertices $w^+,w^-$ for every neighbour $w$, and hence has link degree $2d_\Gamma(v)$.
Suppose the whole link is a cycle. Every link vertex has degree two, so every vertex of $\Gamma$ has degree one. The link is connected, which forces $\Gamma$ to be connected. A connected one-regular graph consists of one edge. Conversely, if $\Gamma$ is one edge, then $S_\Gamma$ is the square <torus> and its unique vertex has link the four-cycle
$$
v^+\ {-}\ w^+\ {-}\ v^-\ {-}\ w^-\ {-}\ v^+.
$$
Thus the vertex link is a cycle exactly when $\Gamma$ is an edge.
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