Some exact results on the Potts model partition function in a magnetic field

We consider the Potts model in a magnetic field on an arbitrary graph G. Using a formula by F Y Wu for the partition function Z of this model as a sum over spanning subgraphs of G, we prove some properties of Z concerning factorization, monotonicity and zeros. A generalization of the Tutte polynomial is presented that corresponds to this partition function. In this context, we formulate and discuss two weighted graph-coloring problems. We also give a general structural result for Z for cyclic strip graphs.