The planted k-factor problem

We consider the problem of recovering an unknown $k$-factor, hidden in a weighted random graph. For $k=1$ this is planted matching problem, while $k=2$ case closely related to travelling salesman problem. The inference solved by exploiting information arising from use two different distributions for weights on edges inside and outside sub-graph. argue that, large size limit, phase transition can appear between full partial recovery as function signal-to-noise ratio. give criterion location transition.