  The Potts model is one of the most popular spin models of statistical
physics. The prevailing majority of work done so far corresponds to the lattice
version of the model. However, many natural or man-made systems are much better
described by the topology of a network. We consider the q-state Potts model on
an uncorrelated scale-free network for which the node-degree distribution
manifests a power-law decay governed by the exponent \lambda. We work within
the mean-field approximation, since for systems on random uncorrelated
scale-free networks this method is known to often give asymptotically exact
results. Depending on particular values of q and \lambda one observes either a
first-order or a second-order phase transition or the system is ordered at any
finite temperature. In a case study, we consider the limit q=1 (percolation)
and find a correspondence between the magnetic exponents and those describing
percolation on a scale-free network. Interestingly, logarithmic corrections to
scaling appear at \lambda=4 in this case.
