Almost Resolvable $4$-Cycle Systems

Abstract

A $4$-cycle system of order $n$ is said to be almost resolvable provided its $4$-cycles can be partitioned into $\frac{n-1}{2}$ almost parallel classes (i.e., $\frac{n-1}{4}$ vertex-disjoint $4$-cycles) and a half parallel class (i.e., $\frac{n-1}{8}$ vertex-disjoint $4$-cycles). We construct an almost resolvable $4$-cycle system of every order $n\equiv 1(\mathrm{mod}8)$ except $9$ (for which no such system exists) and possibly $33,41,$ and $57$.