Internal partitions of $5$-regular and $8$-regular graphs

An \emph{internal partition} of a graph is a partition of its vertex set into two nonempty parts such that every vertex has at least half of its neighbors in its own part. DeVos asked whether, for every fixed positive integer $r$, all but finitely many $r$-regular graphs admit an internal partition. This was previously known to hold only for $r\in \{3,4,6\}$. In this paper, we confirm this for $r\in \{5,8\}$ by proving that every $r$-regular graph on at least ${10}^7$ vertices has an internal partition.