A note on chromatic properties of threshold graphs

In threshold graphs one may find weights for the vertices and a threshold value tt such that for any subset SS of vertices, the sum of the weights is at most the threshold tt if and only if the set SS is a stable (independent) set. In this note we ask a similar question about vertex colorings: given an integer pp, when is it possible to find weights (in general depending on pp) for the vertices and a threshold value tptp such that for any subset SS of vertices the sum of the weights is at most tptp if and only if SS generates a subgraph with chromatic number at most p−1p−1? We show that threshold graphs do have this property and we show that one can even find weights which are valid for all values of pp simultaneously.