Coloring games with multi-located players

In this paper we consider minimum coloring problems with multi-located players, where agents are allowed to occupy different vertices in the conflict graph. The related cooperative games generalize the classical minimum coloring games. We show that minimum coloring games with multi-located players are totally balanced if and only if the related minimum coloring problem is perfect and they are submodular if the underlying graph is complete multi-partite. In the first case, the totally balanced game is a generalized rank game, and in the second case, the submodular game is a (matroid) rank game.