  In 1980, Jackson proved that every 2-connected $k$-regular graph with at most
$3k$ vertices is Hamiltonian. This result has been extended in several papers.
In this note, we determine the minimum number of vertices in a connected
$k$-regular graph that is not Hamiltonian, and we also solve the analogous
problem for Hamiltonian paths. Further, we characterize the smallest connected
$k$-regular graphs without a Hamiltonian cycle.
