Hamiltonian cycles visit every vertex in the graph exactly once (similar to the travelling salesman problem). As a result, neither edges nor vertices can be repeated.

Definition: A graph is considered Hamiltonian if and only if the graph has a cycle containing all of the vertices of the graph. Definition: A Hamiltonian cycle is a cycle that contains all vertices in a graph . If a graph has a Hamiltonian cycle, then the graph is said to be Hamiltonian.

The Hamiltonian number of a connected graph is the length of a Hamiltonian walk . In other words, it is the minimum length of a closed spanning walk in the graph. For a Hamiltonian graph, , where is the vertex count.

Proving a graph has no Hamiltonian cycle [closed]

1. A graph with a vertex of degree one cannot have a Hamilton circuit.
2. Moreover, if a vertex in the graph has degree two, then both edges that are incident with this vertex must be part of any Hamilton circuit.
3. A Hamilton circuit cannot contain a smaller circuit within it.

Hamiltonian cycle is a path in a graph that visits each vertex exactly once and back to starting vertex. This program is to determine if a given graph is a hamiltonian cycle or not. The Java program is successfully compiled and run on a Windows system.

Hamiltonian Path in an undirected graph is a path that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian Path such that there is an edge (in the graph) from the last vertex to the first vertex of the Hamiltonian Path.

In an undirected graph, the Hamiltonian path is a path, that visits each vertex exactly once, and the Hamiltonian cycle or circuit is a Hamiltonian path, that there is an edge from the last vertex to the first vertex.