Independent transversal

For a graph whose vertices are partitioned into parts, an independent transversal is an independent set containing exactly one vertex from each part. If a cycle's parts each have twelve vertices, choose one vertex uniformly and independently per part. A cycle edge joining distinct parts is selected at both ends with probability $1/144$. Each event depends on at most $47$ others, since each part meets at most $24$ cycle edges. The Lovász local lemma applies because $48e/144<1$, proving existence for an arbitrary such partition.