  We study non-centrosymmetric topological superconductivity in correlated
doped quantum spin-Hall insulators (QSHI) on honeycomb lattice without
inversion symmetry where the intrinsic (Kane-Mele) and Rashba spin-orbit
couplings can in general exist. We explore the generic topologically
non-trivial superconducting phase diagram of the model system. Over a certain
parameter space, the parity-mixing superconducting state with co-existing
spin-singlet $d$+$id$ and spin-triplet $p$+$ip$-wave pairing is found. On a
zigzag nanoribbon, the parity-mixing superconducting state shows co-existing
helical and chiral Majorana fermions at edges. Relevance of our results for
experiments is discussed.
