On total coloring of small fullerene nanodiscs

We investigate the total coloring of fullerene nanodiscs, a subclass of cubic planar graphs with girth 5 arising in Chemistry, motivated by a conjecture about the nonexistence of a Type 2 cubic graph of girth at least 5. We prove an auxiliary lemma which says that every central layer of a fullerene nanodisc is 4-total colorable, a necessary condition for the nanodisc to be Type 1, and we contribute by giving 4-total colorings for small fullerene nanodiscs, showing that these graphs are Type 1.