How To Color A Map

We study the maximum differential coloring problem, where an n-vertex graph must be colored with colors numbered 1, 2...n such that the minimal difference between the two colors of any edge is maximized. This problem is motivated by coloring maps in which not all countries are contiguous. Since it is known that this problem is NP-hard for general graphs; we consider planar graphs and subclasses thereof. In Chapter 1 we introduce the topic of this thesis and in Chapter 2 we review relevant definitions and basic results. In Chapter 3 we prove that the maximum differential coloring problem remains NP-hard even for planar graphs. Then, we present tight bounds for regular caterpillars and spider graphs and close-to-optimal differential coloring algorithms for general caterpillars and biconnected triangle-free outer-planar graphs. In Chapter 4 we introduce the (d, kn)-differential coloring problem. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, in this chapter we provide a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. We show that it is NP-complete to determine whether a graph admits a (3, 2n)differential coloring. The same negative result holds for the (b2n/3c, 2n)-differential coloring problem, even when input graph is planar. In Chapter 5 we experimentally evaluate and compare several algorithms for coloring a map. Motivated by different application scenarios, we classify our approaches into two categories, depending on the dimensionality of the underlying color space. To cope with the one dimensional color space (e.g., grayscale colors), we employ the (d, kn)-differential coloring. In Chapter 6 we describe a practical approach for visualizing multiple relationships defined on the same dataset using a geographic map metaphor, where clusters