The cube of a graph $T$, denoted $T^{3}$, is the supergraph of $T$ such that the edge $(u, v)$ is in $T^{3}$ if and only if there is a path between $u$ and $v$ in $T$ with three or fewer edges.
My question is: given a tree $T$, how can we find a hamiltonian path on $T^{3}$