Context. Post-asymptotic giant branch (post-AGB) binaries are surrounded by massive disks of gas and dust that are similar to the protoplanetary disks that are known to surround young stars. Aims. We assembled a catalog of all known Galactic post-AGB binaries featuring disks. We explore the correlations between the different observables with the aim of learning more about potential disk-binary interactions. Methods. We compiled spectral energy distributions of 85 Galactic post-AGB binary systems. We built a color-color diagram to differentiate between the different disk morphologies traced by the characteristics of the infrared excess. We categorized the different disk types and searched for correlations with other observational characteristics of these systems. Results. Between 8 and 12% of our targets are surrounded by transition disks, that is, disks having no or low near-infrared excess. We find a strong link between these transition disks and the depletion of refractory elements seen on the surface of the post-AGB star. We interpret this correlation as evidence of the presence of a mechanism that stimulates the dust and gas separation within the disk and that also produces the transition disk structure. We propose that such a mechanism is likely to be due to a giant planet carving a hole in the disk, effectively trapping the dust in the outer disk parts. We propose two disk evolutionary scenarios, depending on the actual presence of such a giant planet in the disk. Conclusions. We advocate that giant planets can successfully explain the correlation between the transition disks and the depletion of refractory materials observed in post-AGB binaries. If the planetary scenario is confirmed, disks around post-AGB binaries could be a unique laboratory for testing planet-disk interactions and their influence on the late evolution of binary stars. The question of whether such planets are first- or second-generation bodies also remains to be considered. We argue that these disks are ideal for studying planet formation scenarios in an unprecedented parameter space.