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 family_n of triangle-free maximally non-hamiltonian graphs of ordern with |_n| 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.