Maximizing hamiltonian pairs and kk-sets via numerous leaves in a tree

Sharp exponential upper bound, k!n−1k!n−1, on the number of hamiltonian kk-sets (i.e., decompositions into kk hamiltonian cycles) among multigraphs GG is found if the number, nn, of vertices is fixed, n≥3n≥3. Moreover, the upper bound is attained iff G=Cnk where Cnk is the kk-fold nn-cycle CnCn. Furthermore, if G≠Cnk then the number of hamiltonian kk-sets in GG is less than or equal to k!n−1/kk!n−1/k, the bound, if k≥2k≥2, being attained for exactly ⌊n−22⌋ nonisomorphic 2k2k-valent multigraphs GG of order n≥4n≥4. For k≥2k≥2, the number of hamiltonian kk-sets among multigraphs of order at least 3 is even.