  The q-state Potts model can be defined on an arbitrary finite graph, and its
partition function encodes much important information about that graph,
including its chromatic polynomial, flow polynomial and reliability polynomial.
The complex zeros of the Potts partition function are of interest both to
statistical mechanicians and to combinatorists. I give a pedagogical
introduction to all these problems, and then sketch two recent results: (a)
Construction of a countable family of planar graphs whose chromatic zeros are
dense in the whole complex q-plane except possibly for the disc |q-1| < 1. (b)
Proof of a universal upper bound on the q-plane zeros of the chromatic
polynomial (or antiferromagnetic Potts-model partition function) in terms of
the graph's maximum degree.
