On the construction of odd cycle systems

Three obvious necessary conditions for the existence of a k ‐cycle system of order n are that if n > 1 then n ⩾ k, n is odd, and 2 k divides n ( n − 1). We show that if these necessary conditions are sufficient for all n satisfying k ⩽ n < 3 k then they are sufficient for all n. In particular, there exists a 15‐cycle system of order n if and only if n ≡ 1, 15, 21, or 25 (mod 30), and there exists a 21‐cycle system of order n if and only if n ≡ 1, 7, 15, or 21 (mod 42), n ≠ 7. 15.