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To determine whether each graph contains an Eulerian path, Eulerian circuit, Hamiltonian path, or Hamiltonian circuit, let's break down the problem.

### Definitions:
1. **Eulerian Path**: A path that visits every edge of the graph exactly once.
2. **Eulerian Circuit**: A circuit that visits every edge exactly once and returns to the starting vertex. It exists if all vertices have an even degree and the graph is connected.
3. **Hamiltonian Path**: A path that visits every vertex of the graph exactly once.
4. **Hamiltonian Circuit**: A circuit that visits every vertex exactly once and returns to the starting vertex.

### Graph a)
This graph resembles a pentagonal structure with inner edges connecting the vertices.

- **Eulerian Path**: To have an Eulerian path, exactly 0 or 2 vertices must have an odd degree. In this graph, each vertex has an even degree (4), so it contains an Eulerian circuit (and hence, an Eulerian path).
- **Eulerian Circuit**: Yes, because all vertices have an even degree and the graph is connected.
- **Hamiltonian Path**: The graph appears to be fully connected, and we can find a path that visits all vertices exactly once.
- **Hamiltonian Circuit**: Yes, because the graph can visit all vertices exactly once and return to the start.

### Graph b)
This is a star graph with labeled vertices forming a pentagram structure.

- **Eulerian Path**: The degrees of the outer vertices are 2, and the center vertex has a degree of 5 (odd). Since more than two vertices have odd degrees, there is no Eulerian path.
- **Eulerian Circuit**: No, because the central vertex has an odd degree.
- **Hamiltonian Path**: Yes, it is possible to traverse each vertex exactly once.
- **Hamiltonian Circuit**: Yes, because all vertices can be visited exactly once and return to the start.

### Graph c)
This graph resembles a planar rectangular structure with inner connections.

- **Eulerian Path**: Check the degrees: vertices 1, 2, 3, and 4 have degree 3 (odd), while vertices 5, 6, 7, and 8 have degree 2 (even). Since more than two vertices have odd degrees, there is no Eulerian path.
- **Eulerian Circuit**: No, since several vertices have an odd degree.
- **Hamiltonian Path**: Yes, there is a Hamiltonian path that visits all vertices exactly once.
- **Hamiltonian Circuit**: Yes, a circuit exists that visits all vertices exactly once and returns to the starting point.

### Summary:
- **Graph a)**:
  - Eulerian Path: Yes
  - Eulerian Circuit: Yes
  - Hamiltonian Path: Yes
  - Hamiltonian Circuit: Yes
- **Graph b)**:
  - Eulerian Path: No
  - Eulerian Circuit: No
  - Hamiltonian Path: Yes
  - Hamiltonian Circuit: Yes
- **Graph c)**:
  - Eulerian Path: No
  - Eulerian Circuit: No
  - Hamiltonian Path: Yes
  - Hamiltonian Circuit: Yes

Would you like more details on how each criterion was determined?

Here are 5 related questions to explore:
1. How do you determine if a graph is connected?
2. Can a graph have an Eulerian path but not an Eulerian circuit?
3. What is the difference between a Hamiltonian path and an Eulerian path?
4. How does the degree of vertices affect the existence of an Eulerian circuit?
5. Can a graph have multiple Hamiltonian circuits?

**Tip**: A quick way to check for an Eulerian circuit is to verify whether all vertices have even degrees.

Determine whether the following graphs contain an Eulerian path, Eulerian circuit, Hamiltonian path, or Hamiltonian circuit.

Explore whether graphs a, b, and c contain Eulerian paths, Eulerian circuits, Hamiltonian paths, or Hamiltonian circuits. Analyze each graph using Euler's theorem and key principles of graph theory to determine their properties.

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