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The problem asks to determine whether an **Euler path** or an **Euler circuit** exists for each of the given graphs \( G_1 \), \( G_2 \), and \( G_3 \). Let's define these terms before analyzing each graph:

- **Euler Path**: A path in a graph that visits every edge exactly once. An Euler path exists if the graph has exactly two vertices of odd degree (or none if it is a circuit).
- **Euler Circuit**: A circuit (closed path) that visits every edge exactly once. An Euler circuit exists if every vertex in the graph has an even degree.

Now, let’s examine each graph:

### 1. **Graph \( G_1 \)**:
   - **Vertices**: A, B, C, D, E
   - The degree of each vertex:
     - \( \deg(A) = 2 \) (connected to B, C)
     - \( \deg(B) = 3 \) (connected to A, C, D)
     - \( \deg(C) = 3 \) (connected to A, B, D)
     - \( \deg(D) = 3 \) (connected to B, C, E)
     - \( \deg(E) = 1 \) (connected to D)
   
   **Euler Path**: Since there are exactly two vertices of odd degree (B, C, and D), an Euler path exists.
   
   **Euler Circuit**: Since not all vertices have even degrees, there is no Euler circuit.

### 2. **Graph \( G_2 \)**:
   - **Vertices**: A, B, C, D, E, F
   - The degree of each vertex:
     - \( \deg(A) = 4 \)
     - \( \deg(B) = 4 \)
     - \( \deg(C) = 4 \)
     - \( \deg(D) = 4 \)
     - \( \deg(E) = 4 \)
     - \( \deg(F) = 4 \)
   
   **Euler Path**: All vertices have even degrees, so an Euler path exists.

   **Euler Circuit**: Since all vertices have even degrees, an Euler circuit also exists.

### 3. **Graph \( G_3 \)**:
   - **Vertices**: A, B, C, D, E
   - The degree of each vertex:
     - \( \deg(A) = 4 \)
     - \( \deg(B) = 4 \)
     - \( \deg(C) = 4 \)
     - \( \deg(D) = 4 \)
     - \( \deg(E) = 4 \)
   
   **Euler Path**: All vertices have even degrees, so an Euler path exists.

   **Euler Circuit**: Since all vertices have even degrees, an Euler circuit also exists.

### Summary of Results:
- \( G_1 \): Euler Path exists, but no Euler Circuit.
- \( G_2 \): Both Euler Path and Euler Circuit exist.
- \( G_3 \): Both Euler Path and Euler Circuit exist.

Would you like any more details or have questions?

### Related Questions:
1. What is the difference between an Euler path and a Hamiltonian path?
2. Can a graph have more than one Euler circuit?
3. How can you quickly check if a graph has an Euler circuit?
4. What conditions guarantee the existence of an Euler path in a directed graph?
5. How does the concept of degree help in determining graph properties?

### Tip:
A quick way to check for an Euler path or circuit is by counting the vertices with odd degrees—2 for a path, 0 for a circuit!

Find an Euler path or an Euler circuit, if it exists, in each of the following three graphs. If it does not exist, explain why.

Determine whether Euler paths or Euler circuits exist in the three given graphs based on the degree of their vertices. This problem covers graph theory and Euler's theorem, which are essential for understanding paths and circuits in graphs. Suitable for students in Grades 10-12.

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