The AVD-total chromatic number of fullerene molecular graphs

Final version updated according to the journal (DMTCS) requirements, including corrected affiliations and layout adjustments. An \textit{AVD-$k$-total coloring} of a simple graph $G$ is a mapping $\pi:V(G) \cup E(G) \to \{1,\ldots,k\}$, with $k \geq 1$ such that: for each pair of adjacent or incident elements $x,y \in V(G) \cup E(G)$, $\pi(x) \neq \pi(y)$; and for each pair of adjacent vertices $x,y \in V(G)$, sets $\{\pi(x)\} \cup \{\pi(xv) \mid xv \in E(G), v \in V(G)\}$ and $\{\pi(y)\} \cup \{\pi(yv)\mid yv \in E(G), v \in V(G)\}$ are distinct. The \textit{AVD-total chromatic number}, denoted by $\chi''_{a}(G)$ is the smallest $k$ for which $G$ admits an AVD-$k$-total-coloring. We consider a conjecture proposed in 2010 in the thesis of Jonathan Hulgan that any graph~$G$ with maximum vertex degree 3 has $\chi''_{a}(G) \leq 5$. As positive evidence, we prove that several molecular graphs known as fullerene graphs have AVD-total chromatic number equal to 5.