  A total coloring of a graph colors all its elements, vertices and edges, with
no adjacency conflicts. The Total Coloring Conjecture (TCC) is a sixty year old
challenge, says that every graph admits a total coloring with at most maximum
degree plus two colors, and many graph parameters have been studied in
connection with its validity. If a graph admits a total coloring with maximum
degree plus one colors, then it is Type 1, whereas it is Type 2, in case it
does not admit a total coloring with maximum degree plus one colors but it does
satisfy the TCC. Cavicchioli, Murgolo and Ruini proposed in 2003 the hunting
for a Type 2 snark with girth at least 5. Brinkmann, Preissmann and Sasaki in
2015 conjectured that there is no Type 2 cubic graph with girth at least 5. We
investigate the total coloring of fullerene nanodiscs, a class of cubic planar
graphs with girth 5 arising in Chemistry. We prove that the central layer of an
arbitrary fullerene nanodisc is 4-total colorable, a necessary condition for
the nanodisc to be Type 1. We extend the obtained 4-total coloring to a 4-total
coloring of the whole nanodisc, when the radius satisfies r = 5 + 3k, providing
an infinite family of Type 1 nanodiscs.
