Solution to Question 23

What is a Hamiltonian Circuit?

A Hamiltonian Circuit in a graph is a closed loop that visits each vertex exactly once, returning to the starting vertex. The graph should allow this traversal without skipping any vertex or repeating visits to a vertex.

Checking the Given Graph

The graph provided has six vertices: $1, 2, 3, 4, 5, 6$.

To determine if a Hamiltonian Circuit exists:

1. List all vertices: $1, 2, 3, 4, 5, 6$.
2. Check connectivity: The graph is connected, as all vertices are reachable through edges.
3. Find a potential Hamiltonian Circuit: Attempt to find a closed loop that passes through all vertices exactly once.

Potential Circuit:

• Start at vertex $1$.
• Traverse: $1 \to 2 \to 3 \to 5 \to 6 \to 4 \to 1$.

This traversal visits all vertices exactly once and returns to the starting vertex. Therefore, this graph contains a Hamiltonian Circuit.

Justification

The sequence $1 \to 2 \to 3 \to 5 \to 6 \to 4 \to 1$ satisfies:

• Each vertex is visited exactly once.
• The path returns to the starting vertex.
• All edges used are valid edges in the graph.

Thus, the given graph contains a Hamiltonian Circuit.

Would you like further clarification or examples of other paths?

5 Related Questions:

1. What is the difference between a Hamiltonian Circuit and an Euler Circuit?
2. Can a Hamiltonian Circuit exist in a disconnected graph? Why or why not?
3. How can you use algorithms (like backtracking) to determine a Hamiltonian Circuit programmatically?
4. What are some applications of Hamiltonian Circuits in real-world problems, such as logistics or network design?
5. What is Dirac's theorem, and how does it help in identifying Hamiltonian Circuits?

Tip:

To easily identify Hamiltonian Circuits, look for cycles that include all vertices without any duplicates. Paths in sparse graphs often make this clearer.