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(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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Discrete Mathematics

**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.
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.
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