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We prove that, for large $n$, every $3$-connected $D$-regular graph on $n$vertices with $D \geq n/4$ is Hamiltonian. This is best possible and confirms aconjecture posed independently by Bollob\'as and H\"aggkvist in the 1970s. Theproof builds on a structural decomposition result proved recently by the sameauthors.

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