El-Zahar's cycle-cover conjecture

Let $G$ be an $n$-vertex graph, and let $n_1,\dots,n_r\geq 3$ be integers satisfying

$$\sum_{i=1}^{r}n_i=n.$$

El-Zahar's conjecture. If

$$\delta(G) \geq \sum_{i=1}^{r} \lceil n_i/2 \rceil,$$

then $G$ contains $r$ vertex-disjoint cycles with lengths $n_1,\dots,n_r$. This is the classical graph analogue motivating the paper's results on cycle factors in digraphs; the conjecture provides a minimum-degree condition for covering a graph with prescribed vertex-disjoint cycle lengths.