Hamiltonian Cycles
Search for closed routes that visit every vertex once and compare them with edge-covering Eulerian routes.
Free theory articleSearch for closed routes that visit every vertex once and compare them with edge-covering Eulerian routes.
Concept & interpretation
A Hamiltonian cycle visits every vertex exactly once and returns to its start. An Eulerian trail uses every edge exactly once; its existence and construction have efficient algorithms under the appropriate connectivity and degree conditions. Deciding whether a general graph has a Hamiltonian cycle is NP-complete.
Experiment
You can interact with a solver in the Hamiltonian Cycles module.
- Enter a sequence of semicolon-separated edge pairs (e.g.,
0 1; 1 2; 2 3) into the "Edges" input text field. - Click the "Step" button repeatedly to watch the algorithm sequentially test each candidate permutation for valid edges.
- Click the "Finish" button to immediately jump to the end and observe the total Hamiltonian cycle count and the first canonical cycle found in the resulting read-only text fields.