HAMILTONIAN PATH

In graph theory, a Hamiltonian path is a path that visits each vertex exactly once. A Hamiltonian cycle (also called a hamiltonian circuit, vertex tour or graph cycle) is a cycle that visits each vertex exactly once; a graph that contains a Hamiltonian cycle is a Hamiltonian graph. Any Hamiltonian cycle can be converted to a Hamiltonian path by removing one of its edges, but a Hamiltonian path can be extended to Hamiltonian cycle only if its endpoints are adjacent.

The two problems of testing whether a given graph contains a Hamiltonian path and whether it contains a Hamiltonian cycle are both NP-complete. However, certain classes of graphs always contain Hamiltonian paths. For example,it is known that every tournament (a directed graph in which there is exactly one edge connecting each pair of vertices) has an odd number of Hamiltonian paths. Ignoring the start (or end) vertex, a Hamiltonian cycle is a simple path.

History

Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hamilton). Unfortunately, this solution does not generalize to arbitrary graphs.

See also

• Hamiltonian cycle problem
• Eulerian path
• Traveling salesman problem

External links and resources

• Hamiltonian Page : Hamiltonian cycle and path problems, their generalizations and variations
• Weisstein, Eric W., Hamiltonian Circuit. Wolfram Research, 2003.

References

• Rubin, "Frank, "A Search Procedure for Hamilton Paths and Circuits'". Journal of the ACM, Volume 21, Issue 4. October 1974. ISSN 0004-5411
• Nasu, Michiro, "A Study of Hamiltonian Circuit Problem". fourth draft, April 4, 1996 - August 18, 1999
• DeLeon, Melissa, "A Study of Sufficient Conditions for Hamiltonian Cycles". Department of Mathematics and Computer Science, Seton Hall University, South Orange, New Jersey, United States of America. [PDF]
• Hamilton, William Rowan, "Memorandum respecting a new system of roots of unity". Philosophical Magazine, 12 1856
• Hamilton, William Rowan, "Account of the Icosian Calculus". Proceedings of the Royal Irish Academy, 6 1858