https://file.aiquickdraw.com/mathbot/file/173025249354837rphow7.jpg

To determine if this graph has an Euler path or an Euler circuit, we need to check the degrees of each vertex:

1. **Euler Path** exists if a graph has exactly two vertices with an odd degree.
2. **Euler Circuit** exists if all vertices have an even degree.

### Step-by-Step Analysis

From the image:

- Vertex \( A \) has **degree 4**.
- Vertex \( B \) has **degree 3**.
- Vertex \( C \) has **degree 2**.
- Vertex \( D \) has **degree 2**.
- Vertex \( E \) has **degree 3**.

### Conclusion

- **Euler Path**: There are exactly two vertices with odd degrees (vertices \( B \) and \( E \)), so this graph **has an Euler path**.
- **Euler Circuit**: Since not all vertices have an even degree, there is **no Euler circuit**.

This graph has an Euler path but not an Euler circuit.

Would you like more details or have any questions?

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1. What are the conditions for an Euler path and circuit in a graph?
2. How to find the degrees of vertices in a different type of graph?
3. Can a graph have both an Euler path and an Euler circuit?
4. What happens if a graph has more than two vertices of odd degree?
5. How would adding an edge change the Euler properties of a graph?

**Tip:** Always check each vertex's degree to quickly assess the presence of an Euler path or circuit.

Determine whether the graph with vertices A, B, C, D, and E has an Euler path or circuit based on vertex degrees. Learn about Euler's theorem and the conditions required for Euler paths and circuits.

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