Nesting of Cycle Systems of Even Length

Abstract

In this paper, we prove that for any even integer $m\ge 4$, there exists a nested $m$-cycle system of order $n$ if and only if $n\equiv 1\mathrm{mod}2m$, 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 $u\ge 5$, $u\ne 7,8,12,14,18,19,23,24,33,34$.