  A well-known conjecture of Gr\"unbaum and Nash-Williams proposes that
4-connected toroidal graphs are hamiltonian. The corresponding results for
4-connected planar and projective-planar graphs were proved by Tutte and by
Thomas and Yu, respectively, using induction arguments that proved a stronger
result, that every edge is on a hamilton cycle. However, this stronger property
does not hold for 4-connected toroidal graphs: Thomassen constructed
counterexamples. Thus, the standard inductive approach will not work for the
torus. One possible way to modify it is by characterizing the situations where
some edge is not on a hamilton cycle. We provide a contribution in this
direction, by showing that the obvious generalizations of Thomassen's
counterexamples are critical in a certain sense.
