On-line 3-chromatic graphs .2. Critical graphs

On-line coloring of a graph is the following process. The graph is given vertex by vertex (with adjacencies to the previously given vertices) and for the actual vertex a color different from the colors of the neighbors must be irrevocably assigned. The on-line chromatic number of a graph G, chi*(G) is the minimum number of colors needed to color on-line the vertices of G (when it is given in the worst order). A graph G is on-line k-critical if chi*(G)=k, but chi*(G')<k for all proper induced subgraphs G'subset of G. We show that there are finitely many (51) connected on-line 4-critical graphs but infinitely many disconnected ones. This implies that the problem whether chi*(G)less than or equal to 3 is polynomially solvable for connected graphs but leaves open whether this remains true without assuming connectivity. Using the structure descriptions of connected on-line 3-chromatic graphs we obtain one algorithm which colors all on-line 3-chromatic graphs with 4 colors. It is a tight result. This is a companion paper of [1] in which we analyze the structure of triangle-free on-line 3-chromatic graphs.