  Despite the fact that some vertex coloring problems are polynomially solvable
on certain graph classes, most of these problems are not "under control" from a
polyhedral point of view. The equivalence between \emph{optimization} and
\emph{polyhedral separation} suggests that, for these problems, there must
exist formulations admitting some elegant characterization for the polytopes
associated to them. Therefore, it is interesting to study known formulations
for vertex coloring with the goal of finding such characterizations. In this
work we study the asymmetric representatives formulation and we show that the
corresponding coloring polytope, for a given graph $G$, can be interpreted as
the stable set polytope of another graph obtained from $G$. This result allows
us to derive complete characterizations for the corresponding coloring polytope
for some families of graphs, based on known complete characterizations for the
stable set polytope.
