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 $D \geq n/4$ is Hamiltonian. This is best possible and confirms 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.

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