Capstone - 03 - Shortest Path Computation

# Thisfairlyclassicalproblemwassettled

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Unformatted text preview: ring stations (adjacent vertices). Yet, another famous application of graph coloring is “map coloring”. Here our job is to color adjacent states (vertices) on a planar map (e.g., map of the US) with different colors , but use the minimum number of colors. This fairly classical problem was settled by proving that one need no more than 4 colors for such (planar) graphs. 3 The attempt on the left shows a coloring (assignment of guests to tables) that uses 4 colors (i.e., we need 4 tables). The one on the right shows a coloring that uses only 3 colors. One can show that for this graph, three is the minimum number of colors. Try to prove it! Accordingly, we would seat A, B, and E on the first (red) table; D, G, and H on the second (green) table; and C, and F on the third (blue) table. 4 So, how many colors do we need? Well, the answer is relatively easy for some special graphs: 1. For a complete graph with N nodes (a graph where each pair of vertices are connected with an edge), we would obviously need N colors. One can prove this very easily by contradiction – Assume that one can color a complete graph with less than N colors such that no two adjacent vertices have the same color. Since the graph has N vertices, then it must be the case that at least two vertices have the same color. But since there is an edge between any pair of vertices, it follows that two adjacent vertices have the same color, which is a contradiction. 2. For a ring with N nodes (a...
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## This note was uploaded on 02/10/2014 for the course CS 109 taught by Professor Azerbestavros during the Spring '13 term at BU.

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