Rainbow Odd Cycles

We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow cycle (that is, set edges from distinct $O_i$'s, forming an cycle). As part proof, we characterize those families $K_{n+1}$ do not have any cycle. also $K_{n+1}$, as well edge-disjoint nonempty subgraphs without