Monotonic stable solutions for minimum coloring games

For the class of minimum coloring games (introduced by Deng et al. Math Oper Res, 24:751-766, 1999) we investigate the existence of population monotonic allocation schemes (introduced by Sprumont Games Econ Behav 2:378-394, 1990). We show that a minimum coloring game on a graph has a population monotonic allocation scheme if and only if is -free (or, equivalently, if its complement graph is quasi-threshold). Moreover, we provide a procedure that for these graphs always selects an integer population monotonic allocation scheme.