A Graph Coloring Problem with Applications to Data Compression

We study properties of graph colorings that minimize the quantity of color information with respect to a given probability distribution P on the vertices of the graph. The minimum entropy of any coloring of such a probabilistic graph (G,P ) is the chromatic entropy Hχ(G,P ). Applications of the chromatic entropy are found in data compression with side information at the receiver and digital image partition encoding. We show that minimum entropy colorings are hard to compute even if G is planar, a minimum cardinality coloring is given and P is uniform, but that there exists a polynomial algorithm for finding a minimum entropy coloring of complements of triangle-free graphs. We also consider the minimum number of colors χH(G,P ) in a minimum entropy coloring, and show that it is upper-bounded by ∆(G) + 1, where ∆(G) is the maximum degree, and that it is upper-bounded by ∆(G) if P is uniform. Finally, we show that χH(G,P ) can be arbitrarily larger than the chromatic number χ(G) of the graph, even for restricted families of graphs with uniform P .