 Prove that Kr,2r,3r+1 is non-Hamiltonian for any positive integer r

To prove that the complete tripartite graph \( K_{r, 2r, 3r+1} \) is non-Hamiltonian for any positive integer \( r \), we will leverage some properties of bipartite graphs and Hamiltonian cycles.

### 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.

### Proof

We need to show that \( K_{r, 2r, 3r+1} \) does not contain a Hamiltonian cycle. Let's consider the properties and structure of \( K_{r, 2r, 3r+1} \).

1. **Partitioning the Vertex Set**:

   The graph \( K_{r, 2r, 3r+1} \) has three disjoint sets of vertices:
   - \( V_1 \) with \( r \) vertices,
   - \( V_2 \) with \( 2r \) vertices,
   - \( V_3 \) with \( 3r+1 \) vertices.

2. **Total Number of Vertices**:

   The total number of vertices in \( K_{r, 2r, 3r+1} \) is:
   \[
   n = r + 2r + 3r + 1 = 6r + 1.
   \]

3. **Parities of Vertex Sets**:

   Notice that:
   - \( |V_1| = r \) is the size of the first set, which is odd if \( r \) is odd and even if \( r \) is even.
   - \( |V_2| = 2r \) is always even.
   - \( |V_3| = 3r + 1 \) is always odd.

4. **Requirement for Hamiltonian Cycle**:

   For a Hamiltonian cycle to exist in \( K_{r, 2r, 3r+1} \), we need to be able to visit every vertex exactly once in a cyclic order. This implies that we need to alternate between vertices in the different sets such that each vertex is visited exactly once.

5. **Problem with Hamiltonian Cycle in \( K_{r, 2r, 3r+1} \)**:

   Since \( K_{r, 2r, 3r+1} \) has \( 6r+1 \) vertices (which is an odd number), we need to ensure that every set can contribute equally to the cycle in a way that maintains the alternating property required by the Hamiltonian cycle. This means the total number of vertices being odd will make it impossible to alternate correctly between sets due to the imbalance in the parity.

6. **Imbalance of Odd Sets**:

   Given the partitions:
   - \( V_1 \) and \( V_3 \) both contain an odd number of vertices when \( r \) is even (or both even when \( r \) is odd), which creates a parity mismatch that prevents a Hamiltonian cycle from being possible.
   - When traversing an odd number of vertices, we cannot return to the starting vertex while maintaining the strict alternating pattern required for a Hamiltonian cycle, particularly due to the different sizes of \( V_1 \), \( V_2 \), and \( V_3 \).

### Conclusion

Given the structure of \( K_{r, 2r, 3r+1} \), and the requirement for alternating between partitions in a Hamiltonian cycle, the odd total number of vertices ( \( 6r + 1 \) ) leads to an inherent imbalance. This imbalance makes it impossible to form a Hamiltonian cycle, since there would be no way to return to the starting vertex while maintaining the necessary alternating structure.

Thus, \( K_{r, 2r, 3r+1} \) is non-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. What are some examples of Hamiltonian graphs?
2. How can the concept of bipartiteness help in solving other graph theory problems?
3. What are the criteria for a graph to be Hamiltonian?
4. Can you explain the difference between Hamiltonian paths and cycles?
5. What are some real-world applications of Hamiltonian cycles?

Tip: When dealing with Hamiltonian cycles, always consider the parity of the number of vertices and the structure of the graph to determine the feasibility of such cycles.

Explore the proof demonstrating that the complete tripartite graph K_{r, 2r, 3r+1} is non-Hamiltonian for any positive integer r. Understand the structure and properties of tripartite graphs, Hamiltonian cycles, and the implications of vertex partitioning. Suitable for advanced undergraduate students and beyond.

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