Higher-dimensional counterexamples to Hamiltonicity

For $d\ge 2$, we show that all graphs of $d$-polytopes have a Hamiltonian line graph if and only if $d\mathrm{\ne }3$: We exhibit a graph of a $3$-polytope on $252$ vertices whose line graph does not even have Hamiltonian paths. Adapting a construction by Grünbaum and Motzkin, for large $n$ we also construct simple $3$-polytopes on $3n$ vertices in whose line graph any simple path is shorter than $10n^{\alpha }$, for some constant $\alpha <1$. Moreover, we give four elementary counterexamples of plausible extensions to simplicial complexes of four famous results in Hamiltonian graph theory.