Decomposition of \(K_n\) into Cycles of Length At Most Fifty

Abstract

It is shown that the necessary conditions for the existence of a \(k\)-cycle system of order \(n\) are sufficient for \(k \in \{20, 24, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48\}\), thus settling the problem for all \(k \leq 50\).