  We predict two topological superconducting phases in microscopic models
arising from the Berry phase associated with the valley degree of freedom in
gapped Dirac honeycomb systems. The first one is a topological helical
spin-triplet superconductor with a nonzero center-of-mass momentum that does
not break time-reversal symmetry. We also find a topological chiral-triplet
superconductor with Chern number $\pm 1$ with equal-spin-pairing in one valley
and opposite-spin-triplet pairing in the other valley. Our results are obtained
for the Kane-Mele model in which we have explored the effect of three different
interactions, onsite attraction $U$, nearest-neighbor density-density
attraction $V$, and nearest-neighbor antiferromagnetic exchange $J$, within
self-consistent Bogoliubov--de Gennes theory. Transition metal dichalcogenides
and cold atom experiments are promising platforms to explore these phases.
