Monotonic Stable Solutions for Minimum Coloring Games

For the class of minimum coloring games (introduced by Deng et al. (1999)) we investigate the existence of population monotonic allocation schemes (introduced by Sprumont (1990)). We show that a minimum coloring game on a graph G has a population monotonic allocation scheme if and only if G is (P4, 2K2)-free (or, equivalently, if its complement graph G is quasi-threshold). Moreover, we provide a procedure that for these graphs always selects an integer population monotonic allocation scheme.