Cycle Decompositions of the Complete Graph

Abstract

For a positive integer \(n\), let \(G\) be \(K_n\) if \(n\) is odd and \(K_n\) less a one-factor if \(n\) is even. In this paper it is shown that, for non-negative integers \(p\), \(q\) and \(r\), there is a decomposition of \(G\) into \(p\) \(4\)-cycles, \(q\) \(6\)-cycles and \(r\) \(8\)-cycles if \(4p + 6q + 8r = |E(G)|\), \(q = 0\) if \(n < 6\), and \(r = 0\) if \(n < 8\).