  Dwarf carbon (dC) stars, main sequence stars showing carbon molecular bands,
are enriched by mass transfer from a previous asymptotic-giant-branch (AGB)
companion, which has since evolved to a white dwarf. While previous studies
have found radial-velocity variations for large samples of dCs, there are still
relatively few dC orbital periods in the literature and no dC eclipsing
binaries have yet been found. Here, we analyze photometric light curves from
DR5 of the Zwicky Transient Facility for a sample of 944 dC stars. From these
light curves, we identify 34 periodically variable dC stars. Remarkably, of the
periodic dCs, 82\% have periods less than two days. We also provide
spectroscopic follow-up for four of these periodic systems, measuring radial
velocity variations in three of them. Short-period dCs are almost certainly
post-common-envelope binary systems, since the periodicity is most likely
related to the orbital period, with tidally locked rotation and photometric
modulation on the dC either from spots or from ellipsoidal variations. We
discuss evolutionary scenarios that these binaries may have taken to accrete
sufficient C-rich material while avoiding truncation of the thermally pulsing
AGB phase needed to provide such material in the first place. We compare these
dCs to common-envelope models to show that dC stars probably cannot accrete
enough C-rich material during the common-envelope phase, suggesting another
mechanism like wind-Roche lobe overflow is necessary. The periodic dCs in this
paper represent a prime sample for spectroscopic follow-up and for comparison
to future models of wind-Roche lobe overflow mass transfer.
