A bstract In this work we have studied the Kleiss-Kuijf relations for the recently introduced Parke-Taylor factors at one-loop in the CHY approach, that reproduce quadratic Feynman propagators. By doing this, we were able to identify the non-planar one-loop Parke-Taylor factors. In order to check that, in fact, these new factors can describe non-planar amplitudes, we applied them to the bi-adjoint Φ 3 theory. As a byproduct, we found a new type of graphs that we called the non-planar CHY-graphs . These graphs encode all the information for the subleading order at one-loop, and there is not an equivalent of these in the Feynman formalism.