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mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
exactly once. A Hamiltonian cycle (or Hamiltonian circuit) 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. Determining whether such paths and cycles exist in graphs (the
Hamiltonian path problem In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed graph that visits each vertex exactly once) or ...
and
Hamiltonian cycle problem In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed graph that visits each vertex exactly once) or ...
) are
NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...
. Hamiltonian paths and cycles are named after
William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
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 In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
based on
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
with many similarities to the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s (also invented by Hamilton). 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 A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the bo ...
, the
knight's tour A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again im ...
, had been studied in the 9th century in
Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
by Rudrata, and around the same time in
Islamic mathematics Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics ( Aryabhata, Brahmagupta). Important progress was made, such as ...
by al-Adli ar-Rumi. In 18th century Europe, knight's tours were published by
Abraham de Moivre Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He move ...
and
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
.


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 graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...
s'', 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. 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 In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...
with more than two vertices is Hamiltonian * Every
cycle graph In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called ...
is Hamiltonian * Every
tournament A tournament is a competition involving at least three competitors, all participating in a sport or game. More specifically, the term may be used in either of two overlapping senses: # One or more competitions held at a single venue and concentr ...
has an odd number of Hamiltonian paths ( Rédei 1934) * Every
platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
, considered as a graph, is Hamiltonian * The
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...
of a finite Coxeter group is Hamiltonian (For more information on Hamiltonian paths in Cayley graphs, see the
Lovász conjecture In graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in graphs. It says: : Every finite connected vertex-transitive graph contains a Hamiltonian path. Originally László Lovász stated the problem in the opposi ...
.) *
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...
s on nilpotent groups with cyclic commutator subgroup are Hamiltonian. * The
flip graph In mathematics, a flip graph is a graph whose vertices are combinatorial or geometric objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are speci ...
of a convex polygon or equivalently, the rotation graph of
binary tree In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...
s, is Hamiltonian.


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. All Hamiltonian graphs are biconnected, but a biconnected graph need not be Hamiltonian (see, for example, the Petersen graph). An
Eulerian graph In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends ...
(a connected graph in which every vertex has even degree) 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 A tournament is a competition involving at least three competitors, all participating in a sport or game. More specifically, the term may be used in either of two overlapping senses: # One or more competitions held at a single venue and concentr ...
(with more than two vertices) 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 Bondy () is a commune in the northeastern suburbs of Paris, France. It is located from the centre of Paris, in the Seine-Saint-Denis department. In 2019, it had a population of 54,587. Name The name Bondy was recorded for the first time arou ...
Chvátal theorem, which generalizes earlier results by G. A. Dirac (1952) and
Øystein Ore Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics. Life Ore graduated from the University of Oslo in 1922, with ...
. Both Dirac's and Ore's theorems can also be derived from Pósa's theorem (1962). Hamiltonicity has been widely studied with relation to various parameters such as graph
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
,
toughness In materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing.forbidden subgraphs and
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
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 This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B ...
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 graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V a ...
s, 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.


Existence of Hamiltonian cycles in planar graphs


The Hamiltonian cycle polynomial

An algebraic representation of the Hamiltonian cycles of a given weighted digraph (whose arcs are assigned weights from a certain ground field) is the Hamiltonian cycle polynomial of its weighted adjacency matrix defined as the sum of the products of the arc weights of the digraph's Hamiltonian cycles. This polynomial is not identically zero as a function in the arc weights if and only if the digraph is Hamiltonian. The relationship between the computational complexities of computing it and
computing the permanent In linear algebra, the computation of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of the definitions. The permanent is defined ...
was shown by Grigoriy Kogan.


See also

* Barnette's conjecture, an open problem on Hamiltonicity of cubic
bipartite Bipartite may refer to: * 2 (number) * Bipartite (theology), a philosophical term describing the human duality of body and soul * Bipartite graph, in mathematics, a graph in which the vertices are partitioned into two sets and every edge has an en ...
polyhedral graphs * Eulerian path, a path through all edges in a graph *
Fleischner's theorem In graph theory, a branch of mathematics, Fleischner's theorem gives a sufficient condition for a graph to contain a Hamiltonian cycle. It states that, if is a 2-vertex-connected graph, then the square of is Hamiltonian. it is named after He ...
, on Hamiltonian squares of graphs *
Gray code The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representa ...
*
Grinberg's theorem In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is not Hamiltonian. The result has been widely u ...
giving a necessary condition for
planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...
s to have a Hamiltonian cycle *
Hamiltonian path problem In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed graph that visits each vertex exactly once) or ...
, the computational problem of finding Hamiltonian paths *
Hypohamiltonian graph In the mathematical field of graph theory, a graph ''G'' is said to be hypohamiltonian if ''G'' itself does not have a Hamiltonian cycle but every graph formed by removing a single vertex from ''G'' is Hamiltonian. History Hypohamiltonian graphs ...
, a non-Hamiltonian graph in which every vertex-deleted subgraph is Hamiltonian *
Knight's tour A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again im ...
, a Hamiltonian cycle in the knight's graph *
LCF notation In the mathematical field of graph theory, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian cycle. The cycle ...
for Hamiltonian cubic graphs. *
Lovász conjecture In graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in graphs. It says: : Every finite connected vertex-transitive graph contains a Hamiltonian path. Originally László Lovász stated the problem in the opposi ...
that vertex-transitive graphs are Hamiltonian *
Pancyclic graph In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph.. Pancyclic graphs are a generalization of Hami ...
, graphs with cycles of all lengths including a Hamiltonian cycle * Seven Bridges of Königsberg * Shortness exponent, a numerical measure of how far from Hamiltonian the graphs in a family can be *
Snake-in-the-box The snake-in-the-box problem in graph theory and computer science deals with finding a certain kind of path along the edges of a hypercube. This path starts at one corner and travels along the edges to as many corners as it can reach. After it g ...
, the longest induced path in a hypercube * Steinhaus–Johnson–Trotter algorithm for finding a Hamiltonian path in a permutohedron *
Subhamiltonian graph In graph theory and graph drawing, a subhamiltonian graph is a subgraph of a planar Hamiltonian graph.. Definition A graph ''G'' is subhamiltonian if ''G'' is a subgraph of another graph aug(''G'') on the same vertex set, such that aug(''G'') is ...
, a subgraph of a planar Hamiltonian graph * Tait's conjecture (now known false) that 3-regular polyhedral graphs are Hamiltonian * Travelling salesman problem


Notes


References

* *. *. *. *. *. *. *.


External links

* {{MathWorld, title=Hamiltonian Cycle, urlname=HamiltonianCycle
Euler tour and Hamilton cycles
Computational problems in graph theory NP-complete problems Graph theory objects William Rowan Hamilton