On a construction of Thomassen

Using a construction of Thomassen [Discrete Math. 9, 91–96 (1974)] we prove that for infinitely manyn there is a familyℱn of triangle-free maximally non-hamiltonian graphs of ordern with |ℱn| → ∞ exponentially inn. In particular, for everym ≧ 48 we construct such a graph; an infinite number of these provide new “almost extremal” examples in the sense of minimal size.