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4. A student claims to have found an Euler circuit on each graph below as indicated with arrows. a. Name 1 reason why the indicated path is not an Euler circuit. b. Name 2 reasons why the indicated path is not an Euler circuit. 4 8 4 5 HW & quo ad El bloow inbo Amoilling a hert. 5. Determine if the graph contains an Euler circuit. If so, identify an Euler circuit on the graph by number- ing the sequence of edges in the order traveled. If not, explain. abortor B by this doing batonass wil sbr. & to 6. Determine if the graph contains an Euler circuit. If so, identify an Euler circuit on the graph by number- ing the sequence of edges in the order traveled. If not, explain.

4. A student claims to have found an Euler circuit on each graph below as indicated with arrows. a. Name 1 reason why the indicated path is not an Euler circuit. b. Name 2 reasons why the indicated path is not an Euler circuit. 4 8 4 5 HW & quo ad El bloow inbo Amoilling a hert. 5. Determine if the graph contains an Euler circuit. If so, identify an Euler circuit on the graph by number- ing the sequence of edges in the order traveled. If not, explain. abortor B by this doing batonass wil sbr. & to 6. Determine if the graph contains an Euler circuit. If so, identify an Euler circuit on the graph by number- ing the sequence of edges in the order traveled. If not, explain.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Understanding Euler Circuits in Graph Theory This educational resource is designed to help you understand Euler circuits in the context of graph theory. Below, you'll find exercises aiming to test your ability to identify Euler circuits in given graphs. #### Exercise 4: A student claims to have found an Euler circuit on each graph below, indicated with arrows. **a. Examine why the indicated path is not an Euler circuit.** - **Graph Description:** - This graph consists of a simple shape with six vertices, forming a hexagon with diagonals crossing inside. - **Reason:** - To determine if the path is an Euler circuit, check if all vertices of the graph have even degrees. Identify any vertices with odd degrees which invalidate it as an Euler circuit. **b. Examine two reasons why the indicated path is not an Euler circuit.** - **Graph Description:** - The diagram is a combination of a triangle on top of a square, with additional connections forming a house shape. It includes nine edges and four vertices. - **Reasons:** - Identify vertices with odd degrees. - Confirm whether every edge is traveled once without repetition. #### Exercise 5: **Determine if the graph contains an Euler circuit.** - **Graph Description:** - This graph is in the shape of a cube, comprising eight vertices connected with straight lines forming a three-dimensional box. - **Instructions:** - Analyze each vertex's degree. If all vertices have an even degree, there exists an Euler circuit. Number the sequence of edges if Euler circuit exists. If not, explain why. #### Exercise 6: **Determine if the graph contains an Euler circuit.** - **Graph Description:** - This graph consists of a complex pattern of intersections that include multiple triangular connections and a diamond-like pattern inside a rectangular boundary. - **Instructions:** - Similar to Exercise 5, verify the vertices' degrees to check the possibility of an Euler circuit. Order the edges if found. If the circuit does not exist, explain your reasoning. ### Key Concepts - **Euler Circuit:** A circuit that visits every edge of a graph exactly once and returns to the starting vertex. - **Vertex Degree:** The number of edges incident to a vertex. For an Euler circuit to exist, every vertex must have an even degree. Use the principles of Euler circuits to complete these exercises, enhancing your understanding of this fundamental concept in graph theory.
Transcribed Image Text:### Understanding Euler Circuits in Graph Theory This educational resource is designed to help you understand Euler circuits in the context of graph theory. Below, you'll find exercises aiming to test your ability to identify Euler circuits in given graphs. #### Exercise 4: A student claims to have found an Euler circuit on each graph below, indicated with arrows. **a. Examine why the indicated path is not an Euler circuit.** - **Graph Description:** - This graph consists of a simple shape with six vertices, forming a hexagon with diagonals crossing inside. - **Reason:** - To determine if the path is an Euler circuit, check if all vertices of the graph have even degrees. Identify any vertices with odd degrees which invalidate it as an Euler circuit. **b. Examine two reasons why the indicated path is not an Euler circuit.** - **Graph Description:** - The diagram is a combination of a triangle on top of a square, with additional connections forming a house shape. It includes nine edges and four vertices. - **Reasons:** - Identify vertices with odd degrees. - Confirm whether every edge is traveled once without repetition. #### Exercise 5: **Determine if the graph contains an Euler circuit.** - **Graph Description:** - This graph is in the shape of a cube, comprising eight vertices connected with straight lines forming a three-dimensional box. - **Instructions:** - Analyze each vertex's degree. If all vertices have an even degree, there exists an Euler circuit. Number the sequence of edges if Euler circuit exists. If not, explain why. #### Exercise 6: **Determine if the graph contains an Euler circuit.** - **Graph Description:** - This graph consists of a complex pattern of intersections that include multiple triangular connections and a diamond-like pattern inside a rectangular boundary. - **Instructions:** - Similar to Exercise 5, verify the vertices' degrees to check the possibility of an Euler circuit. Order the edges if found. If the circuit does not exist, explain your reasoning. ### Key Concepts - **Euler Circuit:** A circuit that visits every edge of a graph exactly once and returns to the starting vertex. - **Vertex Degree:** The number of edges incident to a vertex. For an Euler circuit to exist, every vertex must have an even degree. Use the principles of Euler circuits to complete these exercises, enhancing your understanding of this fundamental concept in graph theory.
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