 Prove that Kr,2r,3r is Hamiltonian for any positive integer r using Dirac's Theroem


Dirac's Theorem is a powerful tool in graph theory for determining whether a graph is Hamiltonian. The theorem states:

**Dirac's Theorem**: If \(G\) is a graph with \(n \geq 3\) vertices, and every vertex in \(G\) has degree at least \(n/2\), then \(G\) is Hamiltonian.

To apply Dirac's Theorem to \(K_{r, 2r, 3r}\), we need to verify that the degree condition of Dirac's Theorem is satisfied. 

### Steps to Prove \(K_{r, 2r, 3r}\) is Hamiltonian Using Dirac's Theorem

1. **Identify the Number of Vertices**:

   The complete tripartite graph \(K_{r, 2r, 3r}\) has:
   - \(r\) vertices in the first set \(V_1\),
   - \(2r\) vertices in the second set \(V_2\),
   - \(3r\) vertices in the third set \(V_3\).

   Therefore, the total number of vertices \(n\) is:
   \[
   n = r + 2r + 3r = 6r.
   \]

2. **Degree of Each Vertex**:

   In a complete tripartite graph \(K_{r, 2r, 3r}\), each vertex in one set is connected to every vertex in the other two sets. Thus:
   - Each vertex in \(V_1\) is connected to all vertices in \(V_2\) and \(V_3\), giving it a degree of \(2r + 3r = 5r\).
   - Each vertex in \(V_2\) is connected to all vertices in \(V_1\) and \(V_3\), giving it a degree of \(r + 3r = 4r\).
   - Each vertex in \(V_3\) is connected to all vertices in \(V_1\) and \(V_2\), giving it a degree of \(r + 2r = 3r\).

3. **Check Dirac's Condition**:

   According to Dirac's Theorem, we need to show that every vertex in \(K_{r, 2r, 3r}\) has a degree of at least \(n/2\).

   Since \(n = 6r\), we have:
   \[
   \frac{n}{2} = \frac{6r}{2} = 3r.
   \]

   Now, check the degrees:
   - Vertices in \(V_1\) have a degree of \(5r\), which is \( \geq 3r \).
   - Vertices in \(V_2\) have a degree of \(4r\), which is \( \geq 3r \).
   - Vertices in \(V_3\) have a degree of \(3r\), which is \( \geq 3r \).

   Every vertex in \(K_{r, 2r, 3r}\) has a degree of at least \(3r\), satisfying the condition of Dirac's Theorem.

### Conclusion

Since \(K_{r, 2r, 3r}\) meets the degree condition required by Dirac's Theorem (\(\delta(G) \geq \frac{n}{2}\)), we can conclude that \(K_{r, 2r, 3r}\) is Hamiltonian. That is, \(K_{r, 2r, 3r}\) contains a Hamiltonian cycle for any positive integer \(r\).

Would you like more details or have any questions?

Here are 5 questions you might want to ask next:
1. Can you provide an example of constructing a Hamiltonian cycle for \(K_{2, 4, 6}\)?
2. How does Dirac's Theorem relate to other Hamiltonian cycle theorems?
3. Are there exceptions to Dirac's Theorem for certain types of graphs?
4. What are some practical applications of Hamiltonian cycles in computer science?
5. How does one generalize the concept of Hamiltonian cycles to directed graphs?

Tip: When applying Dirac's Theorem, always verify the degree condition first, as it can quickly determine whether the graph is Hamiltonian without needing to construct the cycle explicitly.

Explore the proof that the complete tripartite graph Kr, 2r, 3r is Hamiltonian using Dirac's Theorem in graph theory. Understand the application of Dirac's Theorem and verify the degree conditions for vertices in Kr, 2r, 3r. Suitable for advanced undergraduate students and enthusiasts of graph theory.

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