BDD Techniques for Graph Coloring and Related Problems

Graph coloring is a general solution to the scheduling problem. Variants of the scheduling problem abound in computer engineering CAD problems. BDDs, which have efficient set and data compression properties, are a tool useful for solving NP-Complete problems such as the graph coloring problem. This report explores BDD techniques helpful for determining the chromatic number of a graph, enumerating all valid graph colorings given a set C of colors, and identifying would be odd-cycle causing edges in two-colorable graphs. 1 Introduction The purpose of this report is to describe progress made on a class assigned research project. At the beginning of the term, an original problem was selected. This problem was to find edges, which if added to an existing undirected graph, form odd (even) circuits. Figure 1 shows a possible input graph for this problem. Notice that the original graph contains no odd circuits. To answer the problem, edges b and c would have to be returned. They would form odd circuits if added. Also, edge a, should be returned as a would be even circuit causing edge. a b c Figure 1: Graph input and output for original problem. As with many research project, new problems evolved from the original problem statement. The evolved problems in this report are how to find the chromatic number of an undirected graph and how to find all valid colorings of an undirected graph. Figure 2 shows a graph with a valid coloring with C={green,red,blue}. Since vertices must not be adjacent to vertices with the same color, no valid coloring can be found for this graph with less than three colors. This is not the only valid coloring of this graph using C. For example, vertex c could be blue. Also, vertex a could be red and so on and so forth. For this simple graph, it is easy to tally up 12 valid colorings using C.