  A spatially non-uniform superconducting phase is proposed as the electronic
variational ground state for the attractive interactions between nearest
neighbors on graphene's honeycomb lattice, close to and right at the filling
one half. The state spontaneously breaks the translational invariance of the
lattice into the Kekule pattern of bond order parameters, and it is gapped,
spin triplet, and odd under the sublattice exchange. With the increase of
attractive interactions we first find the transition from the semimetallic
phase into the p-Kekule superconductor, defined as being odd under the exchange
of Dirac points, with the additional discontinuous
superconductor-superconductor transition into the even s-Kekule state, deep
within the superconducting phase. Topological excitations of the Kekule
superconductor and its competition with other superconducting states on the
honeycomb lattice are discussed.
