On-line P-coloring of graphs

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 of generalized trees, we characterize graphs critical for the greedy P-coloring. A class of graphs for which a greedy algorithm always generates optimal P-colorings for the property P = K3-free is given.