Solution to a problem of Bollobás and Häggkvist on Hamilton cycles in regular graphs

We prove that, for large n, every 3-connected D-regular graph on n   vertices with D≥n/4D≥n/4 is Hamiltonian. This is best possible and verifies the only remaining case of a conjecture posed independently by Bollobás and Häggkvist in the 1970s. The proof builds on a structural decomposition result proved recently by the same authors.