(a) Show that if a graph has an Euler tour, then the degree of each of its vertices is even. In the remaining parts, we'll work out the converse: if the degree of every vertex of a connected finite graph is even, then it has an Euler tour. To do this, let's define an Euler walk to be a walk that includes each edge at most once. (b) Suppose that an Euler walk in a connected graph does not include every edge. Explain why there must be an unincluded edge that is incident to a vertex on the walk. In the remaining parts, let w be the longest Euler walk in some finite, connected graph. (c) Show that if w is a closed walk, then it must be an Euler tour. Hint: part (b) (d) Explain why all the edges incident to the end of w must already be in w. (e) Show that if the end of w was not equal to the start of w, then the degree of the end would be odd. Hint: part (d) 14 In some other texts, this is called an Euler circuit.
(a) Show that if a graph has an Euler tour, then the degree of each of its vertices is even. In the remaining parts, we'll work out the converse: if the degree of every vertex of a connected finite graph is even, then it has an Euler tour. To do this, let's define an Euler walk to be a walk that includes each edge at most once. (b) Suppose that an Euler walk in a connected graph does not include every edge. Explain why there must be an unincluded edge that is incident to a vertex on the walk. In the remaining parts, let w be the longest Euler walk in some finite, connected graph. (c) Show that if w is a closed walk, then it must be an Euler tour. Hint: part (b) (d) Explain why all the edges incident to the end of w must already be in w. (e) Show that if the end of w was not equal to the start of w, then the degree of the end would be odd. Hint: part (d) 14 In some other texts, this is called an Euler circuit.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Discrete Mathematics
Transcribed Image Text:**Understanding Euler Tours in Graph Theory**
An **Euler tour** of a graph is a closed walk that includes every edge exactly once. In this discussion, we'll explore a proof of:
**Theorem:** *A connected graph has an Euler tour if and only if every vertex has even degree.*
**Exercise Details:**
(a) **Objective:** Show that if a graph has an Euler tour, then the degree of each of its vertices is even.
(b) **Objective:** Suppose that an Euler walk in a connected graph does not include every edge. Explain why there must be an unincluded edge that is incident to a vertex on the walk.
(c) **Objective:** Given the longest Euler walk, \( w \), in some finite, connected graph, show that if \( w \) is a closed walk, then it must be an Euler tour.
- **Hint:** Refer to part (b)
(d) **Objective:** Explain why all the edges incident to the end of \( w \) must already be in \( w \).
(e) **Objective:** Show that if the end of \( w \) was not equal to the start of \( w \), then the degree of the end would be odd.
- **Hint:** Refer to part (d)
(f) **Conclusion:** Show that if every vertex of a finite, connected graph has even degree, then it has an Euler tour.
*Note: In some other texts, this is called an Euler circuit.*
**Graph Illustration Explanation:**
The image contains a structured breakdown of the theorem's proof, with each part focusing on a specific aspect of the Euler tour in graphs. While the image doesn't contain visual graphs or diagrams, it implicitly analyzes the properties that a connected graph must have to support an Euler tour.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
