What are some examples of graphs with Hamiltonian paths?

Can you give me some examples of graphs that have Hamiltonian paths?

Sure! A Hamiltonian path is a path in a graph that visits each vertex exactly once. Examples include:

1. A simple cycle graph.
2. The complete graph K_n for n ≥ 3.
3. The Petersen graph.

What about the properties of these graphs?

Great question! Hamiltonian graphs often have certain properties, such as:

• They must be connected.
• They can have varying degrees of vertices, but some configurations like complete graphs guarantee Hamiltonian paths.

Are there any algorithms to find Hamiltonian paths?

Yes, there are several algorithms to find Hamiltonian paths, including:

• Backtracking algorithms
• Dynamic programming approaches
• Heuristic methods like genetic algorithms.

Can you explain the backtracking algorithm?

Certainly! The backtracking algorithm explores all possible paths in the graph. It starts from a vertex, marks it as visited, and recursively visits adjacent vertices. If it reaches a dead end, it backtracks and tries another path until it finds a Hamiltonian path or exhausts all options.