The circular chromatic number 
of a finite simple graph 
containing an graph edge is the infimum
of the real numbers 
for which its vertices
can be mapped to a circle of circumference 
so that adjacent vertices
have circular distance at least 1. An edgeless nonempty graph
has 
(Vince 1988, Zhu 2001). Circular distance is the length of the shorter arc between
the two points.
This refines the chromatic number, since 
.
The signed circular chromatic number
extends the definition by allowing positive and negative edges
to impose different distance constraints.
