To determine whether the graphs $G1$ and $G2$ represent an Eulerian circuit, Eulerian path, Hamiltonian circuit, and Hamiltonian path, let's analyze each case systematically.

1. Eulerian Circuit and Eulerian Path

• Conditions for Eulerian Circuit: All vertices must have even degrees.
• Conditions for Eulerian Path: Exactly 0 or 2 vertices can have odd degrees.

Graph $G1$:

• Degrees of vertices:

  • $A: 2, B: 2, C: 3, D: 4, E: 2, F: 2, G: 3$.

• Odd degree vertices: $C$ and $G$ (2 vertices).

  Conclusion:

  • $G1$ does not have an Eulerian circuit (due to odd-degree vertices).
  • $G1$ has an Eulerian path (exactly 2 vertices have odd degrees).

Graph $G2$:

• Degrees of vertices:

  • $a: 2, b: 3, c: 3, d: 4, e: 2, f: 2, g: 3, h: 2$.

• Odd degree vertices: $b, c, g$ (3 vertices).

  Conclusion:

  • $G2$ does not have an Eulerian circuit (odd-degree vertices).
  • $G2$ does not have an Eulerian path (more than 2 odd-degree vertices).

2. Hamiltonian Circuit and Hamiltonian Path

• Conditions for Hamiltonian Circuit: A closed path visits every vertex exactly once and returns to the starting vertex.
• Conditions for Hamiltonian Path: A path visits every vertex exactly once (no need to return to the starting vertex).

Graph $G1$:

• $G1$ has a Hamiltonian path: $A \to B \to E \to G \to F \to D \to C$.
• $G1$ does not have a Hamiltonian circuit because no single closed path covers all vertices.

Graph $G2$:

• $G2$ has a Hamiltonian path: $a \to b \to e \to h \to g \to f \to c \to d$.
• $G2$ does not have a Hamiltonian circuit because no single closed path covers all vertices.

Summary

Graph $G1$:

• Eulerian Circuit: No
• Eulerian Path: Yes
• Hamiltonian Circuit: No
• Hamiltonian Path: Yes

Graph $G2$:

• Eulerian Circuit: No
• Eulerian Path: No
• Hamiltonian Circuit: No
• Hamiltonian Path: Yes

Would you like further clarifications or detailed visualizations for these explanations?

Here are five related questions for deeper understanding:

1. What are the mathematical proofs behind the conditions for Eulerian circuits and paths?
2. How does the degree of vertices affect the existence of Hamiltonian paths?
3. Can we generalize these findings for directed graphs?
4. Are there algorithms to find Eulerian paths or Hamiltonian paths systematically?
5. What are some real-world applications of these graph theories?

Tip: Always start by determining vertex degrees when analyzing graph problems—it simplifies identifying paths and circuits.