The fractal properties of the hexagonal structure of graphene and the graphite intercalation compounds (GIC) are considered. The fractal properties of the single graphite monolayer are described in terms of the compact, perfect and totally disconnected Cantor set. The hexagonal fractal is built using David's Star as the fractal generator. The Hausdorff-Besikovitch fractal dimension is calculated for boundary of the single layer graphene and the GIC graphite monolayer boundary. Also different computational ways are considered for the fractal dimension calculation of the single layer graphene boundary. Comparison is carried out for the graphene boundary fractal with other regular fractals. The fractal dimension of the graphene boundary is shown to be equal to the Koch curve fractal dimension D = ln4/ln3. Thereby, from this point of view the contour boundary of the single graphene layer and the GIC graphite monolayer can be treated as an ideal one-atom-thickness and one dimensional nanoconductor. Such approach may be useful in study of the graphene and graphene's nanostructures. It is shown that this methodology can be expanded for computation of the fractal boundary dimension to the total family 2D crystals of the Kepler-Shubnikov 63net, including hBN, silicene, germanene, etc.