  We derive high-temperature series expansions for the free energy and the
susceptibility of random-bond $q$-state Potts models on hypercubic lattices
using a star-graph expansion technique. This method enables the exact
calculation of quenched disorder averages for arbitrary uncorrelated coupling
distributions. Moreover, we can keep the disorder strength $p$ as well as the
dimension $d$ as symbolic parameters. By applying several series analysis
techniques to the new series expansions, one can scan large regions of the
$(p,d)$ parameter space for any value of $q$. For the bond-diluted 4-state
Potts model in three dimensions, which exhibits a rather strong first-order
phase transition in the undiluted case, we present results for the transition
temperature and the effective critical exponent $\gamma$ as a function of $p$
as obtained from the analysis of susceptibility series up to order 18. A
comparison with recent Monte Carlo data (Chatelain {\em et al.}, Phys. Rev.
E64, 036120(2001)) shows signals for the softening to a second-order transition
at finite disorder strength.
