Given an n-vertex graph Gand two positive integers d, k is an element of N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G(if one exists) with distinct numbers from 1 to kn(treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. 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 (left perpendicular 2n/3 right pendicular, 2n)-differential coloring problem, even in the case where the input graph is planar.

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 ([2n/3], 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., gray-scale 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 of nodes form countries and neighboring countries correspond to nearby clusters. The aim is to provide a visualization that allows us to compare two or more such maps. In the case where we are considering multiple relationships we also provide an interactive tool to visually explore the effect of combining two or more such relationships. Our method ensures good readability and mental map preservation, based on dynamic node placement with node stability, dynamic clustering with cluster stability, and dynamic coloring with color stability. Finally in Chapter 7 we discuss future work and open problems.