  We consider the Potts model in a magnetic field on an arbitrary graph $G$.
Using a formula of 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.
