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 3
n
vertices in whose line graph any simple path is shorter than
10
n
α
, 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.