Nesting of Cycle Systems of Even Length

In this paper, we prove that for any even integer \(m \geq 4\), there exists a nested \(m\)-cycle system of order \(n\) if and only if \(n \equiv 1 \mod{2m}\), with at most \(13\) possible exceptions (for each value of \(m\)). The proof depends on the existence of certain group-divisible designs that are of independent interest. We show that there is a group-divisible design having block sizes from the set \(\{5, 9, 13, 17, 29, 49\}\), and having \(u\) groups of size \(4\), for all \(u \geq 5\), \(u \neq 7, 8, 12, 14, 18, 19, 23, 24, 33, 34\).