On-Line 3-Chromatic Graphs I. Triangle-Free Graphs

Abstract

This is the first half of a two-part paper devoted to on-line 3-colorable graphs. Here on-line 3-colorable triangle-free graphs are characterized by a finite list of forbidden induced subgraphs. The key role in our approach is played by the family of graphs which are both triangle- and (2K2 + K1)-free. Characterization of this family is given by introducing a bipartite modular decomposition concept. This decomposition, combined with the greedy algorithm, culminates in an on-line 3-coloring algorithm for this family. On the other hand, based on the characterization of this family, all 22 forbidden subgraphs of on-line 3-colorable triangle-free graphs are determined. As a corollary, we obtain the 10 forbidden subgraphs of on-line 3-colorable bipartite graphs. The forbidden subgraphs in the finite basis characterization are on-line 4-critical, i.e., they are on-line 4-chromatic but their proper induced subgraphs are on-line 3-colorable. The results of this paper are applied in the companion paper [Discrete Math., 177 (1997), pp. 99--122] to obtain the finite basis characterization of connected on-line 3-colorable graphs (with 51 4-critical subgraphs). However, perhaps surprisingly, connectivity (or the triangle-free property) is essential in a finite basis characterization: there are infinitely many on-line 4-critical graphs.