Is the following conjecture true or false? (Hopefully it is true — I need it as a lemma.)

For every undirected graph $G=(V,E)$ there exist three *pairwise disjoint* sets of vertices $V_1,V_2,V_3$ (whose union is not necessarily $V$) such that for every $i\in\{1,2,3\}$ and for every cycle $C\in G$, $C\cap V_i\neq \emptyset$.

That is, each $V_i$ hits all cycles in $G$, and the $V_i$ are pairwise disjoint.