An asymptotic solution to the cycle decomposition problem for complete graphs

Let 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 = 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.

• Type: Journal
• Year: 2010
• Where: JCT
• Authors: Darryn E. Bryant, Daniel Horsley
• Keywords: Complete Graphs, Edge-disjoint Cycles, JCT 2010, Perfect Matching