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

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