On - Line Graph Coloring and Finite Basis Problems

On-line graph coloring algorithms have been defined in [2] as algorithms whose input graph is presented vertex by vertex, at each step the current vertex is given with all adjacencies to previously given vertices. The algorithm must color irrevocably the current vertex and has to maintain a proper vertex coloring. The simplest and best understood example of an on-line coloring algorithm is the First Fit (or Greedy) algorithm, FF, which assigns the smallest proper color to the current vertex. The chromatic number of a graph G with respect to an on-line coloring algorithm A, XA(G), is the maximum number of colors used by A over all on-line presentations of G. Thus XA(G) measures the worst case behaviour of A on G. The on-line chromatic number of G, X*(G), is the minimum of XA(G) over all on-line algorithms A. Equivalently, the on-line chromatic number can be defined as the value of the following two person game. Drawer and Painter know graph G. Drawer presents the vertices of G one by one and Painter assigns a proper color to the current vertex. Drawer wants to· force as many colors as possible, Painter wants to use as few colors as possible.