  We prove that if an $n$-vertex graph with minimum degree at least $3$
contains a Hamiltonian cycle, then it contains another cycle of length
$n-o(n)$; this implies, in particular, that a well-known conjecture of Sheehan
from 1975 holds asymptotically. Our methods, which combine constructive,
poset-based techniques and non-constructive, parity-based arguments, may be of
independent interest.
