Hamiltonian path
In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. The computational problems of determining whether such paths and cycles exist in graphs are NP-complete; see Hamiltonian path problem for details.
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. This solution does not generalize to arbitrary graphs.
Despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier by Thomas Kirkman, who, in particular, gave an example of a polyhedron without Hamiltonian cycles. Even earlier, Hamiltonian cycles and paths in the knight's graph of the chessboard, the knight's tour, had been studied in the 9th century in Indian mathematics by Rudrata, and around the same time in Islamic mathematics by al-Adli ar-Rumi. In 18th century Europe, knight's tours were published by Abraham de Moivre and Leonhard Euler.
Definitions
A Hamiltonian path or traceable path is a path that visits each vertex of the graph exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices.A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.
Similar notions may be defined for directed graphs, where each edge of a path or cycle can only be traced in a single direction.
A Hamiltonian decomposition is an edge decomposition of a graph into Hamiltonian circuits.
A Hamilton maze is a type of logic puzzle in which the goal is to find the unique Hamiltonian cycle in a given graph.
Examples
- A complete graph with more than two vertices is Hamiltonian
- Every cycle graph is Hamiltonian
- Every tournament has an odd number of Hamiltonian paths
- Every platonic solid, considered as a graph, is Hamiltonian
- The Cayley graph of a finite Coxeter group is Hamiltonian
- Cayley graphs on nilpotent groups with cyclic commutator subgroup are Hamiltonian.
- The flip graph of a convex polygon or equivalently, the rotation graph of binary trees, is Hamiltonian.
Properties
All Hamiltonian graphs are biconnected, but a biconnected graph need not be Hamiltonian.
An Eulerian graph necessarily has an Euler tour, a closed walk passing through each edge of exactly once. This tour corresponds to a Hamiltonian cycle in the line graph, so the line graph of every Eulerian graph is Hamiltonian. Line graphs may have other Hamiltonian cycles that do not correspond to Euler tours, and in particular the line graph of every Hamiltonian graph is itself Hamiltonian, regardless of whether the graph is Eulerian.
A tournament is Hamiltonian if and only if it is strongly connected.
The number of different Hamiltonian cycles in a complete undirected graph on vertices is and in a complete directed graph on vertices is. These counts assume that cycles that are the same apart from their starting point are not counted separately.
Bondy–Chvátal theorem
The best vertex degree characterization of Hamiltonian graphs was provided in 1972 by the Bondy–Chvátal theorem, which generalizes earlier results by G. A. Dirac and Øystein Ore. Both Dirac's and Ore's theorems can also be derived from Pósa's theorem. Hamiltonicity has been widely studied with relation to various parameters such as graph density, toughness, forbidden subgraphs and distance among other parameters. Dirac and Ore's theorems basically state that a graph is Hamiltonian if it has enough edges.The Bondy–Chvátal theorem operates on the closure of a graph with vertices, obtained by repeatedly adding a new edge connecting a nonadjacent pair of vertices and with until no more pairs with this property can be found.
As complete graphs are Hamiltonian, all graphs whose closure is complete are Hamiltonian, which is the content of the following earlier theorems by Dirac and Ore.
The following theorems can be regarded as directed versions:
The number of vertices must be doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph.
The above theorem can only recognize the existence of a Hamiltonian path in a graph and not a Hamiltonian Cycle.
Many of these results have analogues for balanced bipartite graphs, in which the vertex degrees are compared to the number of vertices on a single side of the bipartition rather than the number of vertices in the whole graph.