Hamiltonian Graphs in Discrete Mathematics: Identifying Hamiltonian Cycles and Paths

Explore Hamiltonian graphs and the concept of Hamiltonian cycles and paths. This guide defines Hamiltonian graphs, provides examples of graphs with and without Hamiltonian cycles/paths, and discusses methods for determining their existence.

Hamiltonian Graphs in Discrete Mathematics

What is a Hamiltonian Graph?

A Hamiltonian graph is a connected graph (meaning you can travel between any two vertices by following edges) that contains a Hamiltonian cycle. A Hamiltonian cycle (also called a Hamiltonian circuit) is a cycle (a path that starts and ends at the same vertex) that visits every vertex in the graph exactly once without repeating any edges.

Hamiltonian Paths

A Hamiltonian path is a path that visits every vertex in a graph exactly once, but it doesn't need to return to the starting vertex. It also does not repeat any edges.

Examples: Identifying Hamiltonian Paths

(Three examples illustrating graphs with Hamiltonian paths, and one without, are provided in the original text and should be included here. Each example should clearly show the Hamiltonian path or explain why one does not exist.)

Hamiltonian Circuits

A Hamiltonian circuit (or Hamiltonian cycle) is a closed path that visits every vertex exactly once and returns to its starting point without repeating any edges.

Examples: Identifying Hamiltonian Circuits

(Three examples illustrating graphs with Hamiltonian circuits, and three without, are provided in the original text and should be included here. Each example should clearly show the Hamiltonian circuit or explain why one does not exist.)

Important Points about Hamiltonian Graphs

• Removing a single edge from a Hamiltonian cycle creates a Hamiltonian path.
• If a graph has a Hamiltonian cycle, it also has a Hamiltonian path (but not vice-versa).
• A graph can have multiple Hamiltonian paths and/or cycles.

Examples: Identifying Hamiltonian Graphs

(Six examples of graphs, illustrating those with and without Hamiltonian paths and circuits, are provided in the original text and should be included here. Each example should clearly show a Hamiltonian path and/or circuit (if one exists) or provide a clear explanation of why one does not exist.)

Conclusion

Determining whether a graph is Hamiltonian is a computationally hard problem (NP-complete). While finding Hamiltonian paths and circuits isn't always straightforward, understanding the concept and being able to identify them in simple graphs is essential in graph theory.