  Let $\beta$ be the functor from Set to CHaus which maps each discrete set X
to its Stone-Cech compactification, the set $\beta$ X of ultrafilters on X.
Every graph G with vertex set V naturally gives rise to a graph $\beta G$ on
the set $\beta V$ of ultrafilters on V . In what follows, we interrelate the
properties of G and $\beta G$. Perhaps the most striking result is that G can
be finitely coloured iff $\beta G$ has no loops.
