[Discrete Mathematics] Hamilton Cycles

Summary

In this video, the concept of Hamilton cycles in graph theory is explored, providing a comprehensive explanation. A Hamilton cycle is a cycle that visits every vertex of a graph exactly once. The video demonstrates how to identify Hamilton cycles in various graphs, including hypercubes and bipartite graphs. It also highlights strategies to tackle complex Hamilton cycle problems, such as breaking down larger graphs into simpler components. The importance of bipartite graphs in determining Hamilton cycles is discussed, along with a practical example. The video concludes with tips for tackling these types of problems in discrete mathematics exams.

Takeaways

• 😀 A Hamilton cycle is a cycle in a graph that visits every vertex exactly once and returns to the starting point.
• 😀 A Hamilton cycle requires the graph to have more than two vertices to be possible.
• 😀 In a graph with a Hamilton cycle, no vertex can be repeated, but each must be visited once.
• 😀 An example of a Hamilton cycle is A → B → C → E → F → G → D → A, where each vertex is used exactly once.
• 😀 Finding Hamilton cycles in higher-dimensional hypercubes like Q3 involves specific traversal methods, splitting the graph into parts.
• 😀 For Q3, splitting the graph into Q- and Q-Prime and using reverse traversal in Q2 helps form the Hamilton cycle.
• 😀 The construction of Hamilton cycles can be generalized for higher-dimensional hypercubes like Q4 by splitting and connecting parts in similar ways.
• 😀 Bipartite graphs, where the vertex set is divided into two disjoint sets, cannot have Hamilton cycles if the sets are not equal in size.
• 😀 The key to proving that a bipartite graph cannot have a Hamilton cycle lies in the inability to connect all vertices without reusing edges.
• 😀 A bipartite graph with an equal number of vertices in both sets can potentially have a Hamilton cycle, as demonstrated with a 9-vertex example.
• 😀 Hamilton cycles can be difficult to find in general, but breaking down problems involving bipartite graphs provides an effective strategy.
• 😀 Proofs for Hamilton cycles can be complex, and advanced graph theory courses are better suited for rigorous proofs than introductory discrete math courses.

Q & A

What is a Hamilton cycle in graph theory?

A Hamilton cycle is a cycle in a graph that visits every vertex exactly once and returns to the starting vertex. It requires each vertex to be used once, without repetition.

Can you give an example of a Hamilton cycle?

In the given example, a Hamilton cycle is found in a graph with vertices A, B, C, D, E, F, and G. The cycle goes from A to B to C to E to F to G to D and back to A, using each vertex exactly once.

How do you find a Hamilton cycle in a hypercube?

In a hypercube, you can find a Hamilton cycle by visiting each vertex exactly once. An example is given where the cycle starts at 0,0, and follows the path 0,1,1,1,1,0,0,0, using every vertex without repetition.

What method is used to find a Hamilton cycle in Q3 from Q2?

The method involves splitting the graph into two parts, Q- and Q'-prime. After finding the Hamilton cycle for Q2, the two parts are connected, ensuring each vertex is visited exactly once.

Why are bipartite graphs relevant when discussing Hamilton cycles?

Bipartite graphs are relevant because if the two sets of vertices in a bipartite graph have unequal sizes, it is impossible to form a Hamilton cycle. This is proven by showing that when starting at the larger set, the cycle cannot visit every vertex.

Can a bipartite graph with unequal partition sizes have a Hamilton cycle?

No, a bipartite graph with unequal partition sizes cannot have a Hamilton cycle because you will either run out of vertices to visit or have to reuse an edge or vertex, which violates the conditions of a Hamilton cycle.

What happens when the sizes of the two sets in a bipartite graph are equal?

When the sizes of the two sets are equal in a bipartite graph, a Hamilton cycle is possible because each set can be fully traversed, alternating between the two sets without needing to reuse edges or vertices.

What is the significance of K43 and K7171 in relation to Hamilton cycles?

K43 cannot have a Hamilton cycle because one side has more vertices than the other, while K7171 can have a Hamilton cycle because both sides contain the same number of vertices.

How does the script suggest constructing a Hamilton cycle in a bipartite graph?

The script demonstrates a Hamilton cycle construction by alternating between the two sets of vertices (VA and VB). By ensuring that both sets have the same number of vertices, you can trace a cycle that visits each vertex once.

What does the speaker suggest about proving Hamilton cycles?

The speaker notes that proofs for Hamilton cycles are complex and typically not required in an introductory discrete math course. Instead, the focus is on understanding concepts and using heuristics like bipartite graph analysis.