Online graph coloring with bichromatic exchanges

Greedy algorithms for the graph coloring problem require a large number of colors, even for very simple classes of graphs. For example, any greedy algorithm coloring trees requires $\Omega(\log n)$ colors in the worst case. We consider a variation of the First-Fit algorithm in which the algorithm is allowed to make modifications to previously colored vertices by performing local bichromatic exchanges. We show that such algorithms can be used to find an optimal coloring in the case of bipartite graphs, chordal graphs and outerplanar graphs. We also show that it can find coloring of general planar graphs with $O(\log \Delta)$ colors, where$\Delta$ is the maximum degree of the graph. The question of whether planar graphs can be colored by an online algorithm with bichromatic exchanges using only a constant number of colors is still open. 
This is a joint work with Sylvain Gravier.