On-line -coloring of graphs

For a given induced hereditary property , a -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 . We consider the effectiveness of on-line -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 -coloring algorithm. In the class of generalized trees, we characterize graphs critical for the greedy -coloring. A class of graphs for which a greedy algorithm always generates optimal -colorings for the property = K₃-free is given.