Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 134 / 1 / c / 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-cycleThus the vertex link is a cycle exactly when $\Gamma$ is an edge.