  An internal or friendly partition of a graph is a partition of the vertex set
into two nonempty sets so that every vertex has at least as many neighbours in
its own class as in the other one. It has been shown that apart from finitely
many counterexamples, every 3, 4 or 6-regular graph has an internal partition.
In this note we focus on the $5$-regular case and show that among the subgraphs
of minimum degree at least $3$, there are some which have small intersection.
We also discuss the existence of internal partitions in some families of Cayley
graphs, notably we determine all $5$-regular Abelian Cayley graphs which do not
have an internal partition.
