Questions: A mail carrier needs to deliver mail to each house in the three-block neighborhood shown. He plans to park at one of the street intersections and walk to deliver mail. All streets have houses on both sides. This means that the mail carrier must walk down every street twice, delivering mail to each side separately. b. Choose the correct answer below. A. No, since there four intersections with an odd number of streets going into it, by Euler's theorem there cannot be an Euler circuit. B. No, since Euler's theorem shows that there is an Euler path, but not an Euler circuit for this graph, so the mailman is able to deliver mail to each house without retracing the side of any street, but will not return to the parked truck. C. Yes, since Euler's theorem shows that there is an Euler path, but not an Euler circuit, for this graph, the mailman is able to deliver mail to each house without retracing the side of any street, and return to the parked truck. D. Yes, since there are no odd vertices on the graph there is an Euler circuit, so the mailman can start at a vertex, travel across every edge, and return the same vertex.

A mail carrier needs to deliver mail to each house in the three-block neighborhood shown. He plans to park at one of the street intersections and walk to deliver mail. All streets have houses on both sides. This means that the mail carrier must walk down every street twice, delivering mail to each side separately. b. Choose the correct answer below. A. No, since there four intersections with an odd number of streets going into it, by Euler's theorem there cannot be an Euler circuit. B. No, since Euler's theorem shows that there is an Euler path, but not an Euler circuit for this graph, so the mailman is able to deliver mail to each house without retracing the side of any street, but will not return to the parked truck. C. Yes, since Euler's theorem shows that there is an Euler path, but not an Euler circuit, for this graph, the mailman is able to deliver mail to each house without retracing the side of any street, and return to the parked truck. D. Yes, since there are no odd vertices on the graph there is an Euler circuit, so the mailman can start at a vertex, travel across every edge, and return the same vertex.
Transcript text: A mail carrier needs to deliver mail to each house in the three-block neighborhood shown. He plans to park at one of the street intersections and walk to deliver mail. All streets have houses on both sides. This means that the mail carrier must walk down every street twice, delivering mail to each side separately. b. Choose the correct answer below. A. No, since there four intersections with an odd number of streets going into it, by Euler's theorem there cannot be an Euler circuit. B. No, since Euler's theorem shows that there is an Euler path, but not an Euler circuit for this graph, so the mailman is able to deliver mail to each house without retracing the side of any street, but will not return to the parked truck. C. Yes, since Euler's theorem shows that there is an Euler path, but not an Euler circuit, for this graph, the mailman is able to deliver mail to each house without retracing the side of any street, and return to the parked truck. D. Yes, since there are no odd vertices on the graph there is an Euler circuit, so the mailman can start at a vertex, travel across every edge, and return the same vertex.
failed

Solution

failed
failed

Solution Steps

Step 1: Understand the Problem

The mail carrier needs to deliver mail to every house in a three-block neighborhood. The streets have houses on both sides, so the mail carrier must walk down every street twice (once for each side). The question asks whether the mail carrier can start at a parked truck, deliver mail to every house without retracing any street side, and return to the truck.

Step 2: Apply Euler's Theorem

Euler's theorem states that a graph has an Euler circuit (a path that starts and ends at the same vertex and traverses every edge exactly once) if and only if:

  1. The graph is connected.
  2. Every vertex has an even degree (an even number of edges).

If the graph has exactly two vertices of odd degree, it has an Euler path (a path that starts and ends at different vertices and traverses every edge exactly once) but not an Euler circuit.

Step 3: Analyze the Graph

The problem mentions that there are four intersections with an odd number of streets. This means there are four vertices with odd degrees. According to Euler's theorem:

  • A graph with more than two vertices of odd degree cannot have an Euler circuit or an Euler path.
  • Therefore, the mail carrier cannot deliver mail to every house without retracing some street sides and return to the truck.

Final Answer

The correct answer is A.
\\(\boxed{\text{A. No, since there are four intersections with an odd number of streets going into it, by Euler's theorem there cannot be an Euler circuit.}}\\)

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
33. "Whatever you do, do not sign up for that class. It's really hard!" Yes No
failedAnswered
1. y= 2. r=
failedAnswered
2. Differentiate between negative pressure and positive pressure and what happens in inhalation and exhalation of the lungs.
failedAnswered
Factor the following expression. 24 u^3 w^9+18 u w^3 y^6
failedAnswered
How did Asoka spread Buddhism throughout the Mauryan Empire?