-We study the possible superconducting pairing symmetry mediated by spin and charge fluctuations on the honeycomb lattice using the extended Hubbard model and the random-phaseapproximation method. From 2% to 20% doping levels, a spin-singlet d x 2 −y 2 + idxy-wave is shown to be the leading superconducting pairing symmetry when only the on-site Coulomb interaction U is considered, with the gap function being a mixture of the nearest-neighbor and next-nearestneighbor pairings. When the offset of the energy level between the two sublattices exceeds a critical value, the most favorable pairing is a spin-triplet f -wave which is mainly composed of the next-nearest-neighbor pairing. We show that the next-nearest-neighbor Coulomb interaction V is also in favor of the spin-triplet f -wave pairing.Since the production of graphene (a honeycomb lattice of carbon atoms) in 2004 [1], the realization of superconductivity on the honeycomb lattice have attracted considerable interest [2][3][4][5]. Recently, the studies on the Ca-intercalated bilayer graphene and the graphene laminates observed superconductivities at 4 K [2] and 6.4 K [3] respectively. Furthermore, another recent experimental study also presented evidence for superconductivity in Lidecorated monolayer graphene with the transition temperature around 5.9 K [4]. On the theoretical side, the studies have been extended to models of interacting electrons on the honeycomb lattice, without necessarily concentrating on the parameter regions relevant to graphene, as other systems based on this geometry have been found [6]. Especially, nitrides β-MNCl (M=Hf,Zr) which are composed of alternate stacking of honeycomb layers have been observed to exhibit superconductivity with T c ∼ 15 K for Zr [7] and T c ∼ 25 K for Hf [8] by doping carriers. Various experimental results, including a weak isotope effect [9,10] and the T -linear specific heat [11], have pointed to an unconventional superconducting state, and the magnetic susceptibility measurements [12] suggest that the electron pairings are possibly mediated by magnetic fluctuations in these materials.Many theoretical studies based on the Hubbard model predict a superconducting order parameter with d x 2 −y 2 + id xy symmetry in the spin-singlet channel at half filling and low doping levels [13][14][15][16][17][18][19][20][21][22], while a recent study with the variational cluster approximation and the cellular dynamical mean field theory suggests that the dominant pairing is a spin-triplet with the p x + ip y symmetry [23]. A variational-Monte-Carlo (VMC) study shows that both the d x 2 −y 2 -wave and d x 2 −y 2 + id xy -wave are the possible superconducting pairing symmetry, but the state with d x 2 −y 2 -wave symmetry has the larger condensation energy [24]. Another quantum-Monte-Carlo study predicts that the favored state would have p x + ip y symmetry in the spin-triplet channel but at a large doping level (∼ 80%) [25]. Overall, the pairing symmetry of the possible superconductivity of the interacting electron ...