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

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... ge 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 γ 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 et al., Phys. Rev. E64, 036120(2001)) shows signals for the softening to a second-order transition at finite disorder strength.