An asymptotic solution to the cycle decomposition problem for complete graphs

AbstractLet m1,m2,…,mt be a list of integers. It is shown that there exists an integer N such that for all n⩾N, the complete graph of order n can be decomposed into edge-disjoint cycles of lengths m1,m2,…,mt if and only if n is odd, 3⩽mi⩽n for i=1,2,…,t, and m1+m2+⋯+mt=(n2). In 1981, Alspach conjectured that this result holds for all n, and that a corresponding result also holds for decompositions of complete graphs of even order into cycles and a perfect matching