# Degree reduction step in Dinur's proof of the PCP theorem

In the degree reduction step of Dinur's proof, the input graph $G$ is transformed into a graph $G'$ by replacing each vertex $v \in V(G)$ by a set of vertices, $cloud(v)$, such that $|cloud(v)| = degree_G(v)$, and imposing a degree d expander graph on $cloud(v)$ for all $v \in V(G)$. This makes $G'$ a d+1 regular graph, and the construction ensures that the gap reduces only by a constant factor. I was wondering what would happen if we impose a cycle on each cloud instead? I tried bounding the drop in the gap, but was not able to do so. So, does the proof break down at this step?