Higher-dimensional counterexamples to Hamiltonicity

For d ≥ 2 d ≥ 2 , we show that all graphs of d -polytopes have a Hamiltonian line graph if and only if d ≠ 3 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 3 n vertices in whose line graph any simple path is shorter than 10 nα 10 n α , for some constant α <1 α < 1 . Moreover, we give four elementary counterexamples of plausible extensions to simplicial complexes of four famous results in Hamiltonian graph theory.