The crystallization of alkane melts on carbon nanotubes (CNT) and the surface of graphene nanosheets (GNS) is investigated using molecular dynamics (MD) simulations. The crystallization process of the alkane melts is analyzed in terms of the bond-orientational order parameter, atomic radial distribution for the CNT/alkane, atomic longitudinal distribution for the GNS/alkane, and diffusion properties. The dimensional effects of the different carbon-based nanostructures on the crystallization of alkane melts are shown. It is found that one-dimensional CNT has a stronger ability to induce the crystallization of the polymer than that of two-dimensional GNS, which provides a support at molecular level for the experimental observation [Liet al., J. Am. Chem. Soc., 2006, 128, 1692 and Xu et al., Macromolecules, 2010, 43, 5000]. From the MD simulations, we also find that the crystallization of alkane molecules has been completed with the highly cooperative processes of adsorption and orientation.