  The maximum $k$-colorable subgraph (M$k$CS) problem is to find an induced
$k$-colorable subgraph with maximum cardinality in a given graph. This paper is
an in-depth analysis of the M$k$CS problem that considers various semidefinite
programming relaxations including their theoretical and numerical comparisons.
To simplify these relaxations we exploit the symmetry arising from permuting
the colors, as well as the symmetry of the given graphs when applicable. We
also show how to exploit invariance under permutations of the subsets for other
partition problems and how to use the M$k$CS problem to derive bounds on the
chromatic number of a graph.
  Our numerical results verify that the proposed relaxations provide strong
bounds for the M$k$CS problem, and that those outperform existing bounds for
most of the test instances.
