Minimax relations for the partial q- colorings of a graph

Abstract A partial q-coloring of a graph is a family of q disjoint stable sets, each one representing a “color”; the largest number of colored vertices in a partial q -coloring is a number α q ( G ), extension of the stability number α ( G )= α 1 ( G ). In this note, we investigate the possibilities, for 1⩽ q ⩽ γ ( G ) , to express α q ( G ) by a minimax equality.