Hamiltonian path

This article is about the overall graph theory concept of a Hamiltonian path. For the specific problem of determining whether a Hamiltonian path or cycle exists in a given graph, see Hamiltonian path problem.

A Hamiltonian cycle in a dodecahedron. Like all platonic solids, the dodecahedron is Hamiltonian.

The Herschel graph is the smallest possible polyhedral graph that does not have a Hamiltonian cycle.

In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete.

Hamiltonian paths and cycles and cycle paths 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). This solution does not generalize to arbitrary graphs. However, despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier by Thomas Kirkman.

Definitions

A Hamiltonian path or traceable path is a path that visits each vertex 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 (except for the vertex that is both the start and end, which is visited twice). A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.

Similar notions may be defined for directed graphs, where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices are connected with arrows and the edges traced "tail-to-head").

A Hamiltonian decomposition is an edge decomposition of a graph into Hamiltonian circuits.

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 (Rédei 1934)
• every platonic solid, considered as a graph, is Hamiltonian
• every prism is Hamiltonian^{[citation needed]}
• The Deltoidal hexecontahedron is the only non-hamiltonian Archimedean dual^{[citation needed]}
• Every antiprism is Hamiltonian^{[citation needed]}
• A maximal planar graph with no separating triangles is Hamiltonian.^{[citation needed]}

Properties

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 line graph of a Hamiltonian graph is Hamiltonian. The line graph of an Eulerian graph is Hamiltonian.

A tournament (with more than two vertices) is Hamiltonian if and only if it is strongly connected.

The problem of finding a Hamiltonian cycle may be used as the basis of a zero-knowledge proof.

The number of different Hamiltonian cycles in a complete undirected graph on n vertices is (n − 1)! / 2 and in a complete directed graph on n vertices is (n − 1)!.

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 (1952) and Øystein Ore. In fact, both Dirac's and Ore's theorems are less powerful than what can be derived from Pósa's theorem (1962). Dirac and Ore's theorems basically state that a graph is Hamiltonian if it has enough edges. First we have to define the closure of a graph.

Given a graph G with n vertices, the closure cl(G) is uniquely constructed from G by repeatedly adding a new edge uv connecting a nonadjacent pair of vertices u and v with degree(v) + degree(u) ≥ n until no more pairs with this property can be found.

Bondy–Chvátal theorem

A graph is Hamiltonian if and only if its closure is Hamiltonian.

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.

Dirac (1952)

A simple graph with n vertices (n ≥ 3) is Hamiltonian if every vertex has degree n / 2 or greater.

Ore (1960)

A graph with n vertices (n ≥ 3) is Hamiltonian if, for every pair of non-adjacent vertices, the sum of their degrees is n or greater (see Ore's theorem).

The following theorems can be regarded as directed versions:

Ghouila-Houiri (1960)

A strongly connected simple directed graph with n vertices is Hamiltonian if every vertex has a full degree greater than or equal to n.

Meyniel (1973)

A strongly connected simple directed graph with n vertices is Hamiltonian if the sum of full degrees of every pair of distinct non-adjacent vertices is greater than or equal to 2n − 1.

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.

See also

• Barnette's conjecture, an open problem on Hamiltonicity of cubic bipartite polyhedral graphs
• Eulerian path, a path through all edges in a graph
• Fleischner's theorem, on Hamiltonian squares of graphs
• Grinberg's theorem giving a necessary condition for planar graphs to have a Hamiltonian cycle
• Knight's tour, a Hamiltonian cycle in the knight's graph
• Hamiltonian path problem, the computational problem of finding Hamiltonian paths
• Hypohamiltonian graph, a non-Hamiltonian graph in which every vertex-deleted subgraph is Hamiltonian
• LCF notation for Hamiltonian cubic graphs.
• Lovász conjecture that vertex-transitive graphs are Hamiltonian
• Pancyclic graph, graphs with cycles of all lengths including a Hamiltonian cycle
• Snake-in-the-box, the longest induced path in a hypercube
• Steinhaus–Johnson–Trotter algorithm for finding a Hamiltonian path in a permutohedron
• Tait's conjecture (now known false) that 3-regular polyhedral graphs are Hamiltonian
• Travelling salesman problem

References

• Berge, Claude; Ghouila-Houiri, A. (1962), Programming, games and transportation networks, New York: John Wiley & Sons, Inc. 
• DeLeon, Melissa, "A Study of Sufficient Conditions for Hamiltonian Cycles". Department of Mathematics and Computer Science, Seton Hall University
• Dirac, G. A. (1952), "Some theorems on abstract graphs", Proceedings of the London Mathematical Society, 3rd Ser. 2: 69–81, doi:10.1112/plms/s3-2.1.69, MR 0047308 
• Graham, Ronald L., Handbook of Combinatorics, MIT Press, 1995. ISBN 978-0-262-57170-8.
• 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
• Meyniel, M. (1973), "Une condition suffisante d'existence d'un circuit hamiltonien dans un graphe orienté", Journal of Combinatorial Theory, Ser. B 14 (2): 137–147, doi:10.1016/0095-8956(73)90057-9 
• Ore, O "A Note on Hamiltonian Circuits." American Mathematical Monthly 67, 55, 1960.
• Peterson, Ivars, "The Mathematical Tourist". 1988. W. H. Freeman and Company, NY
• Pósa, L. A theorem concerning hamilton lines. Magyar Tud. Akad. Mat. Kutato Int. Kozl. 7(1962), 225–226.
• Chuzo Iwamoto and Godfried Toussaint, "Finding Hamiltonian circuits in arrangements of Jordan curves is NP-complete," Information Processing Letters, Vol. 52, 1994, pp. 183–189.

External links

• Weisstein, Eric W., "Hamiltonian Cycle", MathWorld.
• Euler tour and Hamilton cycles