Maximal Sets of 2-Factors and Hamiltonian Cycles

In this paper we find, for each integer d, the smallest d-regular graphs which contain no d′-regular subgraphs, 0 > d′ > d. We then find the set of integers Sp2(n) = {m: there exists a maximal set of m edge-disjoint 2-factors of Kn}, as well as SpH2(n) = {m: there exists a maximal set of m edge-disjoint Hamiltonian cycles of Kn}.