  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 $\Phi^3$ 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.
