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Criticality of counterexamples to toroidal edge-hamiltonicity

A well-known conjecture of Grünbaum 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.