Solution to a problem of Bollob\'as and H\"aggkvist on Hamilton cycles in regular graphs

We prove that, for large $n$, every $3$-connected $D$-regular graph on $n$ vertices with $D\ge n\mathrm{/}4$ is Hamiltonian. This is best possible and confirms a conjecture posed independently by Bollob'as and H"aggkvist in the 1970s. The proof builds on a structural decomposition result proved recently by the same authors.