An Euler circuit is a path in a graph that visits every edge exactly once and returns to the starting vertex. For a graph to have an Euler circuit, two key conditions must be met:

1. All vertices must have even degrees (the number of edges connected to each vertex).
2. The graph must be connected, meaning every vertex must be reachable from any other vertex.

Let's analyze each problem in the image based on these conditions:

6. Hexagon Graph (A, B, C, D, E, F):

• Degree of each vertex: All vertices (A, B, C, D, E, F) are connected to two other vertices, giving them a degree of 2.
• Since all vertices have an even degree, and the graph is connected, this is an Euler circuit.

7. Hexagonal Graph with Diagonal (A, B, C, D, E, F):

• Degree of vertices: A, C, D, F have a degree of 2, but B and E have a degree of 3.
• Since B and E have an odd degree, this is not an Euler circuit.

8. Graph with Triangle and Diagonals (A, B, C, D, E, F, G):

• Degrees of vertices: A (3), B (4), C (3), D (3), E (3), F (3), G (6).
• Since multiple vertices (A, C, D, E, F) have an odd degree, this is not an Euler circuit.

9. Oval Shape (A, B, C, D, E):

• Degree of vertices: A (2), B (2), C (2), D (2), E (2).
• All vertices have an even degree, and the graph is connected, so this is an Euler circuit.

10. Star-like Shape (A, B, C, D, E):

• Degree of vertices: A (4), B (3), C (3), D (3), E (3).
• Vertices B, C, D, E have odd degrees, so this is not an Euler circuit.

11. Grid-like Graph (A, B, C, D, E, F, G, H, I):

• Degrees of vertices: A (3), B (3), C (3), D (3), E (4), F (3), G (4), H (3), I (3).
• Multiple vertices (A, B, C, D, F, H, I) have odd degrees, so this is not an Euler circuit.

Conclusion:

• Euler circuits exist in problems 6 and 9.
• For problem 6, you can start at any vertex and follow the edges, for example: A → B → C → D → E → F → A.
• For problem 9, a possible traversal could be: A → B → C → D → E → A.

Do you need more detailed explanations or diagrams of the traversals?

Here are 5 related questions to deepen understanding:

1. What are the specific conditions that make a graph have an Euler path instead of a circuit?
2. Can a graph have an Euler path but not an Euler circuit? Why or why not?
3. How do Euler's circuits differ from Hamiltonian circuits?
4. What happens if only two vertices of a graph have odd degrees?
5. How would removing one edge in problem 6 affect its status as an Euler circuit?

Tip: To find an Euler circuit, always start by checking the degree of each vertex—this often simplifies the analysis significantly!