We study the descriptive complexity of parity games by taking into account the coloring of their game graphs whilst ignoring their ownership structure. Colorings of game graphs are
identified if they determine the same winning regions and strategies, for *all* ownership structures of nodes. The Rabin index of a parity game is the minimum of the maximal color taken over all equivalent coloring functions. We show that deciding whether the Rabin index is at least k is in P for k=1$but NP-hard for all *fixed* k >= 2. We present an
EXPTIME algorithm that computes the Rabin index by simplifying its input coloring function. When replacing simple cycle with cycle detection in that algorithm, its output over-approximates the Rabin index in polynomial time. We evaluate this efficient algorithm as a preprocessor of solvers in detailed experiments: for Zielonka's solver on random and structured parity games and for *partial* solver psolB on random games.