An infinite family of Type 1 fullerene nanodiscs

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Abstract
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.
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