Hamilton paths

Definition

A Hamilton path is a path in a graph that visits each vertex exactly once. It does not need to return to the starting vertex, unlike a Hamiltonian circuit.

5 Must Know Facts For Your Next Test

1. A Hamilton path must visit every vertex of the graph exactly once.
2. Not all graphs contain a Hamilton path; determining its existence can be complex.
3. Finding a Hamilton path is an NP-complete problem, meaning there is no known efficient algorithm to solve it for all cases.
4. If a graph has a Hamiltonian circuit, it also has at least one Hamilton path (by breaking the circuit at any point).
5. The existence of a Hamilton path does not imply the existence of a Hamiltonian circuit.

Review Questions

• What distinguishes a Hamilton path from other types of paths in a graph?
• Why is finding a Hamilton path considered an NP-complete problem?
• Can the existence of a Hamiltonian circuit guarantee the presence of a Hamilton path?

Related terms

Hamiltonian Circuit: A cycle in a graph that visits each vertex exactly once and returns to the starting vertex.

Euler Path: A path that visits every edge of the graph exactly once but does not necessarily visit every vertex.

Graph Theory: The study of graphs and their properties, including paths, circuits, and connectivity.