Hamilton graph

A Hamilton/Hamiltonian graph, or sometimes called a tour is a type of graph that contains a Hamiltonian cycle.

A Hamiltonian cycle is a path that visits each vertex exactly once and returns to the starting vertex, forming a closed cycle.

Hamiltonian graph: a graph that contains a Hamiltonian cycle, visiting each vertex exactly once and returning to the starting vertex.

Rules for Hamilton graphs

• You cannot repeat a vertex (except for the starting/ending vertex).
• You must visit every vertex in the graph exactly once.
• The cycle must return to the starting vertex.
• You can’t repeat edges because that would require revisiting vertices.

Remember that not all graphs contain a Hamiltonian cycle.

Uses of Hamilton graphs

The main use of the Hamiltonian graph concept is in solving routing problems - the common one is called the Traveling Salesman Problem.

Example

Consider the following graph:

    A
  /   \
 B --- C
 |     |
 D --- E

A Hamiltonian cycle in this graph could be: A -> B -> D -> E -> C -> A, because that cycle visits each vertex exactly once and returns to the starting vertex A.

Note that we didn’t use the edge B -> C in this cycle. We don’t need to use all edges in a Hamiltonian cycle; we just need to visit all vertices exactly once!