Within the graph-theoretical SPP (Source-and-Sink Potential) model, a strong omni-conductor can be defined as a molecular graph that is predicted to conduct at the Fermi level, no matter how it is connected by single-atom contacts to the wires. Exhaustive search (n <= 70 for general isomers; n <= 140 for isolated-pentagon isomers) shows that nearly all fullerene isomers C-n in the chemical size range are strong omni-conductors. The isolated-pentagon isomer of C-60 is a strong omniconductor, whereas the isolated-pentagon isomer of C-70, which has a non-bonding LUMO, has some insulating pairs of connections. (C) 2013 Elsevier B.V. All rights reserved.