On the existence of unparalleled even cycle systems

A 2t2t-cycle system of even order vv is a set CC of cycles of length 2t2t whose edges partition the edge-set of Kv−IKv−I (i.e., the complete graph minus the 11-factor II). If v≡0(mod2t), a set of v/2tv/2t vertex-disjoint cycles of CC is a parallel class. If CC has no parallel classes, we call such a system unparalleled.We show that there exists an unparalleled 2t2t-cycle system of order v≡0(mod2t) if and only if v>2t>2v>2t>2.