Almost Resolvable 4-Cycle Systems

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 \( n \equiv 1 \pmod{8} \) except \( 9 \) (for which no such system exists) and possibly \( 33, 41, \) and \( 57 \).