  A complete partition of a graph $G$ is a partition of the vertex set such
that there is at least one edge between any two parts. The largest $r$ such
that $G$ has a complete partition into $r$ parts, each of which is an
independent set, is the achromatic number of $G$. We determine the achromatic
number of polarity graphs coming from generalized polygons. Our colorings of a
family of unitary polarity graphs are used to solve a problem of Axenovich and
Martin on complete partitions of $C_4$-free graphs. Furthermore, these
colorings prove that there are sequences of graphs which are optimally complete
and have unbounded degree, a problem that had been studied for the sequence of
hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch,
and Moorhouse.
