Course 1 Unit 4 Lesson 1 Investigation 2 Making the Circuit

Name: ______Date: ______Pd: ______

Page 254 Your criteria for the optimal sequence for painting the lockers are the defining characteristics of an important property of a graph. An Euler (pronounced oy’lur) circuit is a route through a connected graph such that (1) each edge of the graph is traced exactly once, and (2) the route starts and ends at the same vertex. Given a connected graph, it is often helpful to know if it has an Euler circuit. (The name “Euler” is in recognition of the eighteenth century Swiss mathematician Leonhard Euler. He was the first to study and write about these circuits.)

1. Shown below are graph models of the sidewalks of two sections of a town. Parking meters are placed along these sidewalks.

a. Why would it be helpful for a parking control officer to know if these graphs had Euler circuits?

b. Does the graph model of the east section of town have an Euler circuit? Explain your reasoning.

c. Does the graph model of the west section of town have an Euler circuit? Explain your reasoning.

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2. Graphs a, b, and c below are similar to puzzles enjoyed by people all over the world. In each case, the challenge is to trace the figure. You must trace every edge exactly once without lifting your pencil and return to where you started. That is, the challenge is to trace a Euler circuit through the figure or graph. Place a sheet of paper over each graph and try to trace a Euler circuit. If the graph has a Euler circuit, write down the vertices in order as you trace the circuit.

a. Does this graph have an Euler Circuit? If so, write down the vertices in order as you trace the circuit.

b. Does this graph have an Euler Circuit? If so, write down the vertices in order as you trace the circuit.

c. Does this graph have an Euler Circuit? If so, write down the vertices in order as you trace the circuit.

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3. By looking at the form of an equation you often can predict the shape of the graph of the equation without plotting any points. Similarly, it would be helpful to be able to examine a vertex-edge graph and predict if it has an Euler circuit without trying to trace it. a. Have each member of your group draw a graph with five or more edges that has an Euler circuit. On a separate sheet of paper, draw a graph with five or more edges that does not have an Euler circuit.

b. Sort your group’s graphs into two piles, those that have an Euler circuit and those that do not.

c. Carefully, examine the graphs in the two piles. Describe key ways that graphs with Euler circuits differ from those with no Euler circuit.

d. Try to figure out a way to predict if a graph has a Euler circuit simply by examining its vertices. Check your method of prediction using the graphs from activity 2.

e. Make a conjecture about the properties of a graph that has an Euler circuit. Explain why you think your conjecture is true for any graph with an Euler circuit.

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4. Once you can predict whether a graph has an Euler circuit, it is often still necessary to find the circuit. Consider the graphs below.

a. For each graph, predict whether it has an Euler circuit. i.

ii.

b. Find an Euler circuit in the graph that has one. List the vertices below.

c. Describe the method you used to find your Euler circuit. Describe other possible methods for finding Euler circuits.

5. Skip

6. Choose your preferred method for finding Euler circuits from activity 4. Write specific step-by-step instructions that describe the method you chose. Your instructions should be written so that they apply to any graph, not just the one that you may be working on at the moment. Such a list of step-by-step instructions is called an algorithm.

CHECKPOINT page 257 It is possible to examine a graph and tell if it has an Euler circuit. If it does there are algorithms to finding the circuit.

a. How can you tell if the graph at the right has an Euler circuit without actually tracing it?

Course 1 Unit 4 Lesson 1 Investigation 2 Page 4 of 5 Course 1 Unit 4 Lesson 1 Investigation 2 Making the Circuit b. Find the Euler circuit.

Creating algorithms is an important aspect of mathematics. Two questions should be asked about any algorithm. Does it work? And Is it efficient?

In devising a way to predict whether a graph has an Euler circuit, you probably counted the number of edges at each vertex of the graph. The number of edges touching a vertex is called the degree of the vertex. (If an edge loops back to the same vertex, that counts as two edge touchings.)

ON YOUR OWN page 258

1. For each of the graphs below, check the degree of each vertex and then decide if an Euler circuit exists. If an Euler Circuit exists, use your algorithm to find one.

a.

b.

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