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.