  A circular arc graph is the intersection graph of a collection of connected
arcs on the circle. We solve a Tur'an-type problem for circular arc graphs: for
n arcs, if m and M are the minimum and maximum number of arcs that contain a
common point, what is the maximum number of edges the circular arc graph can
contain? We establish a sharp bound and produce a maximal construction. For a
fixed m, this can be used to show that if the circular arc graph has enough
edges, there must be a point that is covered by at least M arcs. In the case
m=0, we recover results for interval graphs established by Abbott and
Katchalski (1979). We suggest applications to voting situations with interval
or circular political spectra.
