5 The Mathematics of Getting Around 5.1 Euler Circuit Problems 5.2 What Is a Graph? 5.3 Graph Concepts and Terminology 5.4 Graph Models 5.5 Euler’s Theorems 5.6 Fleury’s Algorithm 5.7 Eulerizing Graphs

Modeling One of Euler’s most important insights was the observation that certain types of problems can be conveniently rephrased as graph problems and that, in fact, graphs are just the right tool for describing many real-life situations. The notion of using a mathematical concept to describe and solve a real-life problem is one of the oldest and grandest traditions in mathematics. It is called modeling.

Example 5.15 The Seven Bridges of Königsberg: Part 2 The Königsberg bridges question discussed in Example 5.3 asked whether it was possible to take a stroll through the old city of Königsberg and cross each of the seven bridges once and only once. To answer this question one obviously needs to take a look at the layout of the old city. A stylized map of the city of Königsberg is shown in Fig. 5-13(a).

Example 5.15 The Seven Bridges of Königsberg: Part 2 This map is not entirely accurate–the drawing is not to scale and the exact positions and angles of some of the bridges are changed. Does it matter?

Example 5.15 The Seven Bridges of Königsberg: Part 2 The shape and size of the islands, the width of the river, the lengths of the bridges–none of these things really matter. So, then, what is it that does matter? Surprisingly little.

Example 5.15 The Seven Bridges of Königsberg: Part 2 The only thing that truly matters to the solution of this problem is the relationship between land masses (islands and banks) and bridges. Which land masses are connected to each other and by how many bridges? This information is captured by the red edges in Fig. 5-13(b).

Example 5.15 The Seven Bridges of Königsberg: Part 2 Thus, when we strip the map of all its superfluous information, we end up with the graph model shown in Fig. 5-13(c). The four vertices of the graph represent each of the four land masses; the edges represent the seven bridges.

Example 5.15 The Seven Bridges of Königsberg: Part 2 In this graph an Euler circuit would represent a stroll around the town that crosses each bridge once and ends back at the starting point; an Euler path would represent a stroll that crosses each bridge once but does not return to the starting point.

Example 5.15 Walking the ‘Hood”: Part 2 In Example 5.1 we were introduced to the problem of the security guard who needs to walk the streets of the Sunnyside neighborhood [Fig. 5-14(a)].

Example 5.16 Walking the ‘Hood”: Part 2 The graph in Fig. 5-14(b)–where each edge represents a block of the neighborhood and each vertex an intersection–is a graph model of this problem.

Example 5.16 Walking the ‘Hood”: Part 2 Does the graph have an Euler circuit? An Euler path? Neither? (These are relevant questions that we will learn how to answer in the next section.)

Example 5.17 Delivering the Mail: Part 2 Recall that unlike the security guard, the mail carrier (see Example 5.2) must make two passes through every block that has homes on both sides of the street (she has to physically place the mail in the mailboxes), must make one pass through blocks that have homes on only one side of the street, and does not have to walk along blocks where there are no houses. In this situation an appropriate graph model requires two edges on the blocks that have homes on both

Example 5.17 Delivering the Mail: Part 2 sides of the street, one edge for the blocks that have homes on only one side of the street, and no edges for blocks having no homes on either side of the street. The graph that models this situation is shown in Fig.5-14(c).