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We investigate superconducting order in the extended Hubbard model on thetwo-dimensional graphene lattice using the variational cluster approximation(VCA) with an exact diagonalization solver at zero temperature. Building on theresults of Ref.$[1]$, which identified triplet $p$- and $p+ip$-wavesuperconductivity as the most favored pairing symmetries in that model, weplace uniform SC solutions in competition with the proposed nonuniform Kekul\'e($p+ip$-K) superconducting pattern proposed in Ref.$[2]$. We find that the$p+ip$-K solution is in fact the most favored pairing in most of the phasediagram. In addition, we show that antiferromagnetism can co-exist with the$p+ip$-K state and that both orders are enhanced by their coexistence.

Kekul\'e superconductivity and antiferromagnetism on the graphene lattice