Solving cycle in undirected graph in log space?
The trick is to use Reingold's result that undirected reachability is in logspace. For each vertex $v$, we check whether the connected component containing $v$ contains a cycle by counting the number of vertices and edges. We do that by going over all vertices $u$, check whether $u$ is reachable from $v$, and if so, increase the vertex counter by 1 and the edge counter by half the degree of $u$. After going over all vertices, we have counted the number of vertices $V$ and edges $E$ in the connected component containing $v$. This component contains a cycle if and only if $E \geq V$.