 Prove that Kr,2r,3r is Hamiltonian for any positive integer r


To prove that the complete tripartite graph \(K_{r, 2r, 3r}\) is Hamiltonian for any positive integer \(r\), we need to show that there exists a cycle in \(K_{r, 2r, 3r}\) that visits every vertex exactly once. 

### Definitions and Properties

1. **Complete Tripartite Graph**: A tripartite graph \(K_{a, b, c}\) is a graph whose vertex set can be partitioned into three disjoint sets \(V_1\), \(V_2\), and \(V_3\) with \(|V_1| = a\), \(|V_2| = b\), and \(|V_3| = c\), such that there are no edges within the same set but every possible edge between vertices in different sets is present.

2. **Hamiltonian Cycle**: A graph is Hamiltonian if it contains a cycle that visits each vertex exactly once.

### Strategy

We will construct a Hamiltonian cycle explicitly by showing that we can visit all vertices exactly once and return to the starting vertex, alternating between the sets \(V_1\), \(V_2\), and \(V_3\).

### Construction of the Hamiltonian Cycle

1. **Partitioning the Vertex Set**:
   
   Let \(V_1\), \(V_2\), and \(V_3\) be the disjoint sets of vertices such that:
   - \(|V_1| = r\),
   - \(|V_2| = 2r\),
   - \(|V_3| = 3r\).

2. **Total Number of Vertices**:
   
   The total number of vertices in \(K_{r, 2r, 3r}\) is:
   \[
   n = r + 2r + 3r = 6r.
   \]
   This is an even number, which is conducive to constructing a Hamiltonian cycle.

3. **Constructing the Cycle**:
   
   We will construct a cycle that starts from a vertex in \(V_1\), alternates between vertices in \(V_2\) and \(V_3\), and returns to the starting vertex. The alternation ensures that we use all vertices from each set exactly once.

### Detailed Construction Steps

1. **Start from a Vertex in \(V_1\)**:
   
   Let \(v_1 \in V_1\) be the starting vertex.

2. **Alternate Between \(V_2\) and \(V_3\)**:
   
   From \(v_1\), move to a vertex in \(V_2\), say \(v_2\). From \(v_2\), move to a vertex in \(V_3\), say \(v_3\). Continue this pattern, alternating between \(V_2\) and \(V_3\).

   Since \(V_2\) has \(2r\) vertices and \(V_3\) has \(3r\) vertices, we can continue this pattern until all vertices in \(V_2\) and \(V_3\) are visited.

3. **Return to \(V_1\)**:
   
   After visiting the last vertex in \(V_3\), return to a new vertex in \(V_1\), and repeat the alternating pattern until all vertices in \(V_1\) are visited.

### Ensuring All Vertices are Visited

Since the total number of vertices is \(6r\), and we alternate between sets, we will visit:

- Each vertex in \(V_1\) once (total \(r\) vertices),
- Each vertex in \(V_2\) twice in sequence (total \(2r\) vertices),
- Each vertex in \(V_3\) thrice in sequence (total \(3r\) vertices).

This alternating pattern will ensure that we cover all vertices exactly once.

### Conclusion

Given the even total number of vertices and the ability to alternate between the disjoint sets \(V_1\), \(V_2\), and \(V_3\), we can construct a Hamiltonian cycle for \(K_{r, 2r, 3r}\) that visits every vertex exactly once and returns to the starting vertex. Thus, \(K_{r, 2r, 3r}\) is Hamiltonian 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 show an example of constructing a Hamiltonian cycle for a specific \(K_{r, 2r, 3r}\)?
2. What are some other types of graphs that are always Hamiltonian?
3. How do you determine if a general graph is Hamiltonian?
4. Are there algorithms for finding Hamiltonian cycles in graphs?
5. What is the significance of Hamiltonian cycles in practical applications?

Tip: When constructing Hamiltonian cycles, breaking down the problem into smaller components and ensuring an even distribution of vertices can help simplify the construction process.

Explore the proof that the complete tripartite graph Kr, 2r, 3r is Hamiltonian for any positive integer r. Understand the concepts of complete graphs, Hamiltonian cycles, and the application of the Hamiltonian Cycle Theorem. Suitable for advanced high school students and beyond.

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