For \(d \ge 2\) , we show that all graphs of d-polytopes have a Hamiltonian line graph if and only if \(d \ne 3\) : 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 \(10 n^{\alpha }\) , 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.
