  The fractional and circular chromatic numbers are the two most studied
non-integral refinements of the chromatic number of a graph. Starting from the
definition of a coloring base of a graph, which originated in work related to
ergodic theory, we formalize the notion of a gyrocoloring of a graph: the
vertices are colored by translates of a single Borel set in the circle group,
and neighbouring vertices receive disjoint translates. The corresponding
gyrochromatic number of a graph always lies between the fractional chromatic
number and the circular chromatic number. We investigate basic properties of
gyrocolorings. In particular, we construct examples of graphs whose
gyrochromatic number is strictly between the fractional chromatic number and
the circular chromatic number. We also establish several equivalent definitions
of the gyrochromatic number, including a version involving all finite abelian
groups.
