Hamiltonicity in connected regular graphs

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.