On the non-optimality of four color coding of image partitions

Recent interest in region based image coding has given rise to graph coloring based partition encoding methods. These methods are based on the four color theorem for planar graphs, and assume that a coloring for a graph with the minimum possible number of colors will result in the most compressible representation. In this paper we show that this assumption is wrong. We show that there exist graphs with chromatic number k that can be colored with k+1 colors resulting in bitmaps representing image partitions which are more compressible than the corresponding bitmaps generated using any k coloring of the same graph. We conclude with some conjectures on optimal coloring of weighted graphs.