"Let $G$ be a planar graph. Show that every pair of vertex-disjoint odd cycles in $G^c$ is connected by an edge." Can't figure out why "odd" matters.

Question

If $C_1,C_2$ are vertex-disjoint cycles in $G^c$, of lengths $m,n$ respectively, not connected by an edge, then their complement has a $K_{m,n}$ minor with $m,n\geq 3$, so $G$ contains $K_{3,3}$ as a minor and hence cannot be planar. But I must be missing something here since the problem assumes that the $C_1,C_2$ are odd cycles; not just any old cycles. What am I missing?