  We analytically calculate the nuclear-spin interactions of a single electron
confined to a carbon nanotube or graphene quantum dot. While the
conduction-band states in graphene are p-type, the accordant states in a carbon
nanotube are sp-hybridized due to curvature. This leads to an interesting
interplay between isotropic and anisotropic hyperfine interactions. By using
only analytical methods, we are able to show how the interaction strength
depends on important physical parameters, such as curvature and isotope
abundances. We show that for the investigated carbon structures, the 13C
hyperfine coupling strength is less than 1 mu-eV, and that the associated
electron-spin decoherence time can be expected to be several tens of
microseconds or longer, depending on the abundance of spin-carrying 13C nuclei.
Furthermore, we find that the hyperfine-induced Knight shift is highly
anisotropic, both in graphene and in nanotubes of arbitrary chirality.
