Let $G = (V, E)$ be a finite, simple, undirected graph with $V \cap E = \emptyset$. The *total graph* $T(G)$ is defined on the vertex set $V \cup E$ and its edge set is given by $$E(T(G)) = E \cup \big\{\{e, f\}: e, f \in E\text{ and } |e\cap f| = 1\big\}\cup \big\{\{v, e\}: v\in V, e\in E, v\in e\big\}.$$

**Question.** If there is a Hamiltonian path in $G$, is there a Hamiltonian path in $T(G)$?