An asymptotic solution to the cycle decomposition problem for complete graphs

Let m 1 , m 2 , … , m t 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 m 1 , m 2 , … , m t if and only if n is odd, 3 ⩽ m i ⩽ n for i = 1 , 2 , … , t , and m 1 + m 2 + ⋯ + m t = ( n 2 ) . 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.