Abstract
For d ≥ 2, we show that all graphs of d-polytopes have a Hamiltonian line graph if and
only if d ̸= 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 10nα, for some constant α < 1. Moreover, we give
four elementary counterexamples of plausible extensions to simplicial complexes of
four famous results in Hamiltonian graph theory.