Got pretty strange question in the HW:
$G$ is a connected, simple graph with $|E(G)|$ even. I need to prove that there exist a partition of edges into pairwise disjoint pairs, where each pair is path of length 2. This must be done, using Tutte's theorem for perfect matching, somehow.
The only way I can see it can be done, is by contracting paths of length 2 into one edge, thus creating a graph, that will be more acceptable for Tutte's theorem. I can't see a clear way to do that, though.
Any clue will be highly appreciated.