Diagonally switchable 4-cycle systems

A diagonally switchable 4-cycle system of order n, briefly DS4CS(n), is a 4-cycle system in which by replacing each 4-cycle (a, b, c, d) covering pairs ab, bc, cd, da by either of the 4-cycles (a, c, b, d) or (a, b, d, c) another 4-cycle system is obtained. We prove that a DS4CS(n) exists if and only if n ≡ 1 (mod 8), n ≥ 17 with the possible exception of n = 17. AMS classification: 05B30