A new class of chromatic, or potential, polynomial is considered. These polynomials are obtained by restricting the potential differences on adjacent vertices to either odd or even parity. For colourings involving an odd number of colours, the polynomials can be evaluated to give an indicator for directed cuts on rooted graphs. Corresponding flow polynomials give directed path indicators. It is shown how the polynomials arising from these restricted colourings and flows can be used to generate the correlation function for directed interaction models which generalize the standard Potts model. A further statistical mechanical model is introduced where odd and even potential differences are given different weights. In this case when each vertex has an odd number of states the percolation limit of this model is Redner’s oriented diode model.