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Path in a graph that visits each vertex exactly once
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_path
Problem of finding a cycle through all vertices of a graph
The Hamiltonian path problem is a topic discussed in the fields of complexity theory and graph theory. It decides if a directed or undirected graph, G
Hamiltonian_path_problem
Topics referred to by the same term
Look up Hamiltonian in Wiktionary, the free dictionary. Hamiltonian may refer to: Hamiltonian mechanics, a formalism based on: Hamiltonian (mechanics)
Hamiltonian
path if and only if there is a Hamiltonian path in G. The Hamiltonian path problem is NP-complete, and hence the minimum path cover problem is NP-hard. However
Path_cover
Sequence of edges which join a sequence of vertices on a given graph
includes every vertex of the graph without repeats is known as a Hamiltonian path. Two paths are vertex-independent (alternatively, internally disjoint or
Path_(graph_theory)
Decomposition of a graph into hamiltonion cycles
mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions
Hamiltonian_decomposition
Cubic graph with 10 vertices and 15 edges
The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. It is the smallest bridgeless cubic graph with no Hamiltonian cycle. It is hypohamiltonian
Petersen_graph
Node ordering for directed acyclic graphs
directed Hamiltonian path in the DAG. If a Hamiltonian path exists, the topological sort order is unique; no other order respects the edges of the path. Conversely
Topological_sorting
Problem of finding the longest simple path for a given graph
critical path in scheduling problems. The NP-hardness of the unweighted longest path problem can be shown using a reduction from the Hamiltonian path problem:
Longest_path_problem
Mathematical problem set on a chessboard
general Hamiltonian path problem in graph theory. The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem
Knight's_tour
NP-hard problem in combinatorial optimization
find a Hamiltonian cycle with the least weight. This is more general than the Hamiltonian path problem, which only asks if a Hamiltonian path (or cycle)
Travelling_salesman_problem
mechanics Hamiltonian optics Hamiltonian principle, see Hamilton's principle Hamiltonian system Hamiltonian vector field In mathematics: Hamiltonian path, in
List of things named after William Rowan Hamilton
List_of_things_named_after_William_Rowan_Hamilton
Problem in graph theory
a Hamiltonian path? More unsolved problems in mathematics In graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in
Lovász_conjecture
Directed graph where each vertex pair has one arc
finite number n {\displaystyle n} of vertices contains a Hamiltonian path, i.e., directed path on all n {\displaystyle n} vertices (Rédei 1934). This is
Tournament_(graph_theory)
String in combinatorial math
+ 12 = 12312 = 312. Any Hamiltonian path through the created graph is a superpermutation, and the problem of finding the path with the smallest weight
Superpermutation
Trail in a graph that visits each edge once
odd-degree vertices Hamiltonian path – a path that visits each vertex exactly once. Route inspection problem, search for the shortest path that visits all
Eulerian_path
In mathematics, the Hamiltonian cycle polynomial of an n×n-matrix is a polynomial in its entries, defined as ham ( A ) = ∑ σ ∈ H n ∏ i = 1 n a i , σ
Hamiltonian_cycle_polynomial
Sequence of locally optimal choices
Yuster refers to as the greedy proof that every tournament contains a Hamiltonian path. Since there is no formal definition of what a greedy algorithm is
Greedy_algorithm
Catalan solid with 60 faces
an Archimedean solid. It is one of six Catalan solids to not have a Hamiltonian path among its vertices. It is topologically identical to the nonconvex
Deltoidal_hexecontahedron
On degree sums and Hamiltonian cycles
would create at least one new Hamiltonian cycle, and the edges other than xy in such a cycle must form a Hamiltonian path v1v2...vn in H with x = v1 and
Ore's_theorem
Non-commutative algebraic structure
Hamilton's work in this area resulted indirectly in the terms Hamiltonian circuit and Hamiltonian path in graph theory. He also invented the icosian game as a
Icosian_calculus
Formulation of quantum mechanics
enters the path integrals (for interactions of a certain type, these are coordinate-space, or Feynman path integrals), than the Hamiltonian. Possible downsides
Path-integral_formulation
Computing using molecular biology hardware
proof-of-concept use of DNA as a form of computation which solved the seven-point Hamiltonian path problem. Since the initial Adleman experiments, advances have occurred
DNA_computing
Concept in graph theory
Tutte paths is their close relationship to Hamiltonian paths and cycles, paths and cycles in a graph that visit every vertex exactly once. A Tutte path is
Tutte_path
On Hamiltonian cycles in toroidal graphs
toroidal graph has a Hamiltonian cycle, and (with W. Zang) that every 4-vertex-connected toroidal graph has a Hamiltonian path. Kawarabayashi, Ken-ichi;
Grünbaum–Nash-Williams conjecture
Grünbaum–Nash-Williams_conjecture
Adding edges to make a graph Hamiltonian
to make them Hamiltonian. Wu, Q. S.; Lu, Chin Lung; Lee, Richard C. T. (2000), "An approximate algorithm for the weighted Hamiltonian path completion problem
Hamiltonian_completion
Graphs formed by a hypercube's edges and vertices
{\displaystyle n>1} has a Hamiltonian cycle, a cycle that visits each vertex exactly once. Additionally, a Hamiltonian path exists between two vertices
Hypercube_graph
Shortest Path First Flooding algorithm Route inspection problem Hamiltonian path Hamiltonian path problem Knight's tour Traveling salesman problem Nearest neighbour
List_of_graph_theory_topics
Sampling algorithm
The Hamiltonian Monte Carlo algorithm (originally known as hybrid Monte Carlo) is a Markov chain Monte Carlo method for obtaining a sequence of random
Hamiltonian_Monte_Carlo
Type of graph in graph theory
graphs that do not contain a Hamiltonian path but such that every subset of n − 1 vertices may be connected by a path. Analogous definitions of hypohamiltonicity
Hypohamiltonian_graph
Graph with all path lengths between each two vertices
Panconnected graphs are also a generalization of Hamiltonian-connected graphs (graphs that have a Hamiltonian path connecting every pair of vertices). Several
Panconnectivity
Planar bipartite graph with 25 vertices and 31 edges
whose neighbour has degree 3 is removed, the resulting graph has no Hamiltonian path. This property was used by Tutte when combining three Walther graphs
Walther_graph
Theorem on Hamiltonian graphs
contain a Hamiltonian cycle. It states that, if G {\displaystyle G} is a biconnected graph, then the square of G {\displaystyle G} is Hamiltonian. It is
Fleischner's_theorem
Computer that uses photons or light waves
and an oscilloscope. The first problem attacked in this way was the Hamiltonian path problem. The simplest problem is the subset sum problem. An optical
Optical_computing
Bipartite non-Hamiltonian polyhedral graph
polyhedron), and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices. The polyhedron whose
Herschel_graph
Disproven graph theory
Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by
Tait's_conjecture
Topics referred to by the same term
Hamiltonian mechanics Hamilton–Jacobi equation, a set of physical equations in Hamiltonian mechanics Hamiltonian (quantum mechanics) Hamiltonian path
Hamilton
Tree graph with all nodes within distance 1 from central path
line graph of an arbitrary tree so that it contains a Hamiltonian path (the size of its Hamiltonian completion) equals the minimum number of edge-disjoint
Caterpillar_tree
never less than the chromatic number. Hamiltonian A Hamiltonian path or Hamiltonian cycle is a simple spanning path or simple spanning cycle: it covers
Glossary_of_graph_theory
On Hamiltonian cycles in planar graphs
W. T. Tutte states that every 4-vertex-connected planar graph has a Hamiltonian cycle. It strengthens an earlier theorem of Hassler Whitney according
Tutte's theorem on Hamiltonian cycles
Tutte's_theorem_on_Hamiltonian_cycles
Formulation of classical mechanics using momenta
physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics
Hamiltonian_mechanics
Cycle through all length-k sequences
The de Bruijn sequences can be constructed by taking a Hamiltonian path of an n-dimensional de Bruijn graph over k symbols (or equivalently
De_Bruijn_sequence
Hamiltonian completion Hamiltonian path problem, directed and undirected. Induced subgraph isomorphism problem Graph intersection number Longest path
List_of_NP-complete_problems
Generalization of depth-first search trees
context of infinite graphs. All depth-first search trees and all Hamiltonian paths are Trémaux trees. In finite graphs, every Trémaux tree is a depth-first
Trémaux_tree
Classic problem in graph theory
featured in different charity events. Eulerian path Five room puzzle Glossary of graph theory Hamiltonian path Icosian game Travelling salesman problem Three
Seven_Bridges_of_Königsberg
Card game produced by Mattel
a way to play all cards in a hand is equivalent to searching for a Hamiltonian path on a graph with vertices representing each card, and edges connecting
Uno_(card_game)
Topological quantum field theory
Kulshreshtha, D.S.; Mueller-Kirsten, H. J. W.; Vary, J. P. (2009). "Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under
Chern–Simons_theory
Form of computing using molecular biology
this model to solve a few NP-complete problems. Specifically, the hamiltonian path problem (HPP) and some versions of the set cover problem are a few
Peptide_computing
Japanese peg solitaire variant
3-satisfiability, or by a parsimonious reduction from the closely related Hamiltonian path problem. Andersson, Daniel (2007), "HIROIMONO Is NP-Complete", in Crescenzi
Goishi_Hiroi
Sliding puzzle with fifteen pieces and one space
(1999) gave another proof, based on defining equivalence classes via a Hamiltonian path. Wilson (1974) studied the generalization of the 15 puzzle to arbitrary
15_puzzle
Representation of cubic graphs
Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian cycle. The cycle itself includes two out of the three adjacencies for
LCF_notation
Game of finding cycles on a dodecahedron
by Irish mathematician William Rowan Hamilton. It involves finding a Hamiltonian cycle on a dodecahedron, a polygon using edges of the dodecahedron that
Icosian_game
integral of a smooth path of Hamiltonian vector fields Yt. Vladimir Arnold conjectured that the number of fixed points of a generic Hamiltonian diffeomorphism
Spectral_invariants
Variant of the traveling salesman problem
in discrete or combinatorial optimization. The problem is to find the Hamiltonian cycle (visiting each node exactly once) in a weighted graph which minimizes
Bottleneck traveling salesman problem
Bottleneck_traveling_salesman_problem
Hamiltonian coloring, named after William Rowan Hamilton, is a type of graph coloring. Hamiltonian coloring uses a concept called detour distance between
Hamiltonian_coloring
Kind of binary decision diagram
the longest such paths are Hamiltonian, with a size of 2,707,075. ZDDs in this case, are efficient for simple paths and Hamiltonian paths. Define 64 input
Zero-suppressed decision diagram
Zero-suppressed_decision_diagram
Shared independent set of two matroids
from the Hamiltonian path problem in directed graphs. Given a directed graph G with n vertices, and specified nodes s and t, the Hamiltonian path problem
Matroid_intersection
Edges that hit all cycles in a graph
has a Hamiltonian path, and the Hamiltonian paths correspond one-for-one with minimal feedback arc sets, disjoint from the corresponding path. The Hamiltonian
Feedback_arc_set
edges as to specify a Hamiltonian decomposition (a decomposition into Hamiltonian paths), then those edges also form a Hamiltonian Decomposition in H {\displaystyle
Graph_amalgamation
Ordering of binary values, used for positioning and error correction
n ( i ) {\displaystyle Q_{n}(i)} . A monotonic Gray code is then a Hamiltonian path in Q n {\displaystyle Q_{n}} such that whenever δ 1 ∈ E n ( i ) {\displaystyle
Gray_code
Rod-shaped, gram-negative bacterium
program E. coli to solve complicated mathematics problems, such as the Hamiltonian path problem. A computer to control protein production of E. coli within
Escherichia_coli
Indian theoretical physicist
theory models, string theory models and D-brane actions using the Hamiltonian, path integral and BRST quantization methods, constrained dynamics, construction
Usha_Kulshreshtha
Unsolved problem in graph theory
Unsolved problem in mathematics Is every cubic bipartite polyhedral graph Hamiltonian? More unsolved problems in mathematics Barnette's conjecture is an unsolved
Barnette's_conjecture
non-Hamiltonian polyhedron, by putting together three such fragments. The "compulsory" edges of the fragments, that must be part of any Hamiltonian path through
Tutte_graph
View of quantum mechanics
interaction picture is a special case of unitary transformation applied to the Hamiltonian and state vectors. Haag's theorem says that the interaction picture doesn't
Interaction_picture
numerical parameter of a family of graphs that measures how far from Hamiltonian the graphs in the family can be. Intuitively, if e {\displaystyle e}
Shortness_exponent
Inherent difficulty of computational problems
algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines
Computational complexity theory
Computational_complexity_theory
Graph containing cycles of all possible lengths
number of vertices in the graph. Pancyclic graphs are a generalization of Hamiltonian graphs, graphs which have a cycle of the maximum possible length. An
Pancyclic_graph
Overview of and topical guide to algorithms
algorithm Graph coloring Clique problem Independent set (graph theory) Hamiltonian path problem Travelling salesman problem String-searching algorithm Knuth–Morris–Pratt
Outline_of_algorithms
Family of graphs based on the Fibonacci sequence
beginning with a 1 bit). Every Fibonacci cube has a Hamiltonian path. More specifically, there exists a path that obeys the partition described above: it visits
Fibonacci_cube
Class of undirected graphs defined from systems of sets
vertices forms the endpoints of a Hamiltonian path in the graph. In particular this means that it has a Hamiltonian cycle. It is also known that the Johnson
Johnson_graph
polynomial magnitude. In particular, there is a reduction from the Hamiltonian path problem, on an n {\displaystyle n} -vertex unweighted graph G {\displaystyle
Zero-weight_cycle_problem
Type of spanning tree
(Garey & Johnson 1979). This can be shown by a reduction from the Hamiltonian path problem. It remains NP-complete even if k is fixed to a value ≥ 2.
Degree-constrained spanning tree
Degree-constrained_spanning_tree
On Hamiltonian cycles in planar graphs
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
Grinberg's_theorem
Intersection graph of unit intervals on the real line
to solve the shortest path problem, and to construct Hamiltonian paths and maximum matchings, all in linear time. A Hamiltonian cycle can be found from
Indifference_graph
Complexity class
decision problems. Boolean satisfiability problem (SAT) Knapsack problem Hamiltonian path problem Travelling salesman problem (decision version) Subgraph isomorphism
NP-completeness
Irish mathematician and physicist (1805–1865)
game or Hamilton's puzzle in 1856. It is based on the concept of a Hamiltonian path in graph theory. In 1824, Hamilton was introduced at Edgeworthstown
William_Rowan_Hamilton
Problem in quantum information science
Hamiltonian simulation (also referred to as quantum simulation) is a problem in quantum information science that attempts to find the computational complexity
Hamiltonian_simulation
Every graph has evenly many odd vertices
the Hamiltonian paths in G {\displaystyle G} beginning at u {\displaystyle u} and continuing through edge u v {\displaystyle uv} . Two such paths p 1
Handshaking_lemma
Two-player board game
NP-hard. This is proven by a reduction from the problem of finding the Hamiltonian path of a cubic subgraph of the square grid graph. Generalized Amazons (that
Game_of_the_Amazons
Area of discrete mathematics
theorem proving and modeling the elaboration of linguistic structure. Hamiltonian path problem Minimum spanning tree Route inspection problem (also called
Graph_theory
However, if no such Hamiltonian path exists, then the best traveling salesman tour must have weight at least |V|. Thus, Hamiltonian Path reduces to |V|/(|V|-1)-gap
Gap_reduction
Key result in Hamiltonian mechanics and statistical mechanics
mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Function used in optimal control theory
The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. It can be understood as an instantaneous increment of
Hamiltonian_(control_theory)
Proving validity without revealing other data
she knows a Hamiltonian cycle in H, then she translates her Hamiltonian cycle in G onto H and only uncovers the edges on the Hamiltonian cycle. That is
Zero-knowledge_proof
Cubic graph with 28 vertices and 42 edges
conjecture asks for an Hamiltonian path and is verified by the Coxeter graph. Only five examples of vertex-transitive graph with no Hamiltonian cycles are known :
Coxeter_graph
Subgraph of planar graph with Hamiltonian cycle
and graph drawing, a subhamiltonian graph is a subgraph of a planar Hamiltonian graph. A graph G is subhamiltonian if G is a subgraph of another graph
Subhamiltonian_graph
Irish contributions to science, technology, and engineering
graph theory exploring paths along the edges of a dodecahedron. His "Icosian game" challenged players to find a Hamiltonian path, a problem that remains
Timeline of Irish inventions and discoveries
Timeline_of_Irish_inventions_and_discoveries
Physics experiment
atoms and molecules. The experiment belongs to a general class of "double path" experiments, in which two diffracted waves reconverge, creating an interference
Double-slit_experiment
Graph theory concept
is NP-hard. This can be shown by constructing a reduction from the Hamiltonian path problem. For directed graphs, finding the minimum degree spanning tree
Minimum_degree_spanning_tree
Discipline in genetics
assembly, Eulerian path strategies, and overlap-layout-consensus (OLC) strategies. OLC strategies ultimately try to create a Hamiltonian path through an overlap
Genomics
Overview of mechanics based on the least action principle
and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space)
Analytical_mechanics
Tree which includes all vertices of a graph
the spanning tree with the fewest leaves (closely related to the Hamiltonian path problem), the minimum-diameter spanning tree, and the minimum dilation
Spanning_tree
Two special graphs in graph theory
graph with 56 vertices and 84 edges, named after Felix Klein. It is Hamiltonian, has chromatic number 3, chromatic index 3, radius 6, diameter 6 and
Klein_graphs
Product of geometric length and refractive index
{\textstyle \Lambda } is the optical path length of C {\textstyle C} . Air mass (astronomy) Lagrangian optics Hamiltonian optics Fermat's principle Optical
Optical_path_length
Algorithm characteristic in computations
Gurevich, Yuri; Shelah, Saharon (1987), "Expected computation time for Hamiltonian path problem", SIAM Journal on Computing, 16 (3): 486–502, doi:10.1137/0216034
Average-case_complexity
Class of quantum field theory models
4249/scholarpedia.8508. Kulshreshtha, U.; Kulshreshtha, D. S. (2002). "Front-Form Hamiltonian, Path Integral, and BRST Formulations of the Nonlinear Sigma Model". International
Non-linear_sigma_model
1987 American TV series or program
Parade") Displacement of fluids ("The Problem of the Trojan Hamburger") Hamiltonian path ("The Case of the Smart Dummy") Process of elimination ("The Case of
Mathnet
Relationship between branches of physics
equation with the path integral formulation of quantum mechanics using a simple nonrelativistic one-dimensional single-particle Hamiltonian composed of kinetic
Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
Relation_between_Schrödinger's_equation_and_the_path_integral_formulation_of_quantum_mechanics
Hamiltonian operator for molecules
molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei
Molecular_Hamiltonian
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