An infinite family of Type 1 fullerene nanodiscs

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.

Figures (21)

• Figure 1: The smallest nanodiscs $D_{2}$ and $D_{3}$.
• Figure 2: Vertices $I,D,E,F,J$ force the existence of a Large Forbidden Face.
• Figure 3: Chain of unbalanced hexagons contained in a sequence of layers of $D_r$, $r \geq 3$ starting from the central layer $L_c$ with an unbalanced hexagon.
• Figure 4: Chain of unbalanced hexagons contained in consecutive layers of $D_r$, $r \geq 3$ starting from the central layer $L_c$ until we reach a Large Forbidden Face.
• Figure 5: (a) Choice of $u_0$ and $v_0$ in $C_{12r-6}$ and $C_{12r-6}^{*}$, respectively; (b) Case 1; (c) Case 2.

Theorems & Definitions (22)

• Theorem 1: Yap, 1996 yap1996
• Conjecture 2: Brinkmann, Preissmann and Sasaki, $2015$ Brinkmann
• Lemma 1: Large Forbidden Face
• proof
• Lemma 2: Balanced Hexagons
• proof
• Lemma 3: Alternating Pentagons
• proof
• Lemma 4
• proof