6th Edition

ISBN: 9780134462356

Author: Pirnot

Publisher: PEARSON

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Chapter 4.1, Problem 38E

Represent the maps given in Exercises 37-40 by graphs as we did in Example 6. Recall that we join two vertices by an edge if and only if the states that they represent share a stretch of common border

Chapter 4.1, Problem 38E, Represent the maps given in Exercises 37-40 by graphs as we did in Example 6. Recall that we join , example 1

Chapter 4.1, Problem 38E, Represent the maps given in Exercises 37-40 by graphs as we did in Example 6. Recall that we join , example 2

Example 6 Solving the Four-Color Problem for South America

Model the map of South America by a graph and use this graph to color the map using at most four colors.

Solution: In this problem, we have a set of countries, some of which are related in that they share a common border. Therefore, we can model this situation by a graph.

We will represent each country by a vertex; if two countries share a common border, we draw an edge between the corresponding vertices. This graph appears in Figure 4.17 .

Chapter 4.1, Problem 38E, Represent the maps given in Exercises 37-40 by graphs as we did in Example 6. Recall that we join , example 3

Note that we connect the vertices representing Peru and Colombia with an edge because they share a common boundary. We do not connect the vertices representing Argentina and Peru, because they have no boundary in common.

We can rephrase the map-coloring question now as follows: Using four or fewer colors, can we color the vertices of a graph so that no two vertices of the same edge receive the same color? It is easier to think about coloring a graph than it is to think about coloring the original map.

We show one coloring using four colors in Figure 4.18 and another coloring that I generated on my iPad using a graph theory app called Graphynx. Notice that Graphynx again had to use four colors to color the graph.

Chapter 4.1, Problem 38E, Represent the maps given in Exercises 37-40 by graphs as we did in Example 6. Recall that we join , example 4

Figure 4.18 Coloring of graph of South America.

Chapter 4.1, Problem 38E, Represent the maps given in Exercises 37-40 by graphs as we did in Example 6. Recall that we join , example 5

Graphynx coloring of graph of South America.

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Chapter 4.1, Problem 37E

Chapter 4.1, Problem 39E

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Solution Summary: The author explains how to join two vertices by an edge if the states that they represent share a stretch of common border.

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