https://mathoverflow.net/questions/16393/finding-a-cycle-of-fixed-length I like this solution the best specially for 4 length:)
Also phys wizard says u have to do O(V^2). I believe that we need only O(V)/O(V+E).
If the graph is connected then DFS will visit all nodes. If the graph has connected sub graphs then each time we run a DFS on a vertex of this sub graph we will find the connected vertices and wont have to consider these for the next run of the DFS. Therefore the possibility of running for each vertex is incorrect.
