Maximally non-hamiltonian graphs of girth 7

We describe a sufficient condition for graphs used in a construction of Thomassen (which yields hypohamiltonian graphs) to produce maximally non-hamiltonian (MNH) graphs as well. Then we show that the Coxeter graph fulfils this sufficient condition, and thus applying the Thomassen’s construction to multiple copies of the Coxeter graph yields infinitely many MNH graphs with girth 7. So far, the Coxeter graph was the only known example of a MNH graph of girth 7; also no MNH graph of girth greater than 7 has been found yet. Finally, the Isaacs’ flower snarks Jk for odd k ≥ 5 are shown to fulfil (for certain vertices) this sufficient condition as well.