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For a given induced hereditary property P, a P-coloring of a graph G is an assignment of one color to each vertex such that the subgraphs induced by each of the color classes have property P. We consider the effectiveness of on-line P-coloring algorithms and give the generalizations and extensions of selected results known for on-line proper coloring algorithms. We prove a linear lower bound for the performance guarantee function of any stingy on-line P-coloring algorithm. In the class ofdoi:10.7151/dmgt.1331 fatcat:kgtohsrapbei3g4xwb6rqkiwku