  We investigated the thermal conductivity K of graphene ribbons and graphite
slabs as the function of their lateral dimensions. Our theoretical model
considered the anharmonic three-phonon processes to the second-order and
included the angle-dependent phonon scattering from the ribbon edges. It was
found that the long mean free path of the long-wavelength acoustic phonons in
graphene can lead to an unusual non-monotonic dependence of the thermal
conductivity on the length L of a ribbon. The effect is pronounced for the
ribbons with the smooth edges (specularity parameter p>0.5). Our results also
suggest that - contrary to what was previously thought - the bulk-like 3D
phonons in graphite can make a rather substantial contribution to its in-plane
thermal conductivity. The Umklapp-limited thermal conductivity of graphite
slabs scales, for L below ~ 10 micrometers, as log(L) while for larger L, the
thermal conductivity approaches a finite value following the dependence K_0 -
A\timesL^-1/2, where K_0 and A are parameters independent of the length. Our
theoretical results clarify the scaling of the phonon thermal conductivity with
the lateral sizes in graphene and graphite. The revealed anomalous dependence
K(L) for the micrometer-size graphene ribbons can account for some of the
discrepancy in reported experimental data for graphene.
