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Topic: Hamiltonian cycle

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	The Hamiltonian Page
		Hamiltonian cycles avoiding the arcs of prescribed subtournaments, by J. Bang-Jensen, G. Gutin, and A. Yeo.
		Hamilton Cycles in Random Graphs and Digraphs, by Colin Cooper and Alan Frieze.
		Hamiltonian paths and cycles in hypertournaments, by G. Gutin and A.
	alife.ccp14.ac.uk /memetic/~moscato/Hamilton.html (1318 words)

	Hamiltonian Cycles in the Vertex-Adjacency Dual
		A Hamiltonian cycle is a walk in a graph G that starts from a vertex v of G and ends at v after visiting all the vertices of G exactly once.
		The Hamiltonian cycle is a special case of the Traveling Salesman Problem (known as the TSP problem), where given a number of cities and the costs of travelling from one to the other, a salesman needs to find the cheapest roundtrip route that visits each city and returns to the starting point.
		Hamiltonian cycles in triangulations are studied when adjacency is defined as vertex-adjacency, where two triangles are considered to be adjacent if they share exactly at least one vertex.
	cgm.cs.mcgill.ca /~perouz/cs507/vertexdual/introduction.htm (1100 words)

	The Hamiltonian Page
		Hamilton Cycles in Random Graphs and Digraphs, by Colin Cooper and Alan Frieze.
		Hamiltonian cycles avoiding the arcs of prescribed subtournaments, by J. Bang-Jensen, G. Gutin, and A. Yeo.
		Hamiltonian paths and cycles in hypertournaments, by G. Gutin and A.
	www.ing.unlp.edu.ar /cetad/mos/Hamilton.html (1335 words)

	Hamiltonian path - Wikipedia, the free encyclopedia
		A Hamiltonian cycle (or Hamiltonian circuit) is a cycle in an undirected graph which visits each vertex exactly once and also returns to the starting vertex.
		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.
		A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once (except the vertex which is both the start and end, and so is visited twice).
	en.wikipedia.org /wiki/Hamiltonian_cycle (659 words)

	Hamiltonian path problem - Wikipedia, the free encyclopedia
		In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph (whether directed or undirected).
		The Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G.
		The Hamiltonian cycle problem is a special case of the traveling salesman problem, obtained by setting the distance between two cities to a finite constant if they are adjacent and infinity otherwise.
	en.wikipedia.org /wiki/Hamiltonian_path_problem (357 words)

	PlanetMath: Hamiltonian graph
		is Hamiltonian if it has a Hamiltonian cycle.
		A useful condition both necessary and sufficient for a graph to be Hamiltonian is not known.
		This is version 8 of Hamiltonian graph, born on 2001-10-24, modified 2006-10-23.
	planetmath.org /encyclopedia/Hamiltonian.html (118 words)

	PCGrate -> Hamiltonian Cycle Problem
		It is based on the concept of successive iterative decreases in cycle length, in which the iterative procedure begins with a randomly chosen cycle.
		As different edges of the graph are assigned the weights 0 and 1, the cycle length is determined in view of the edges with the unit weight.
		Polynomial complexity of the algorithm results from the fact that the number of iterations for each unit edge of the cycle (i.e., when a correlation is made between the unit edges of the cycle and all zero edges of the graph) is finite.
	www.pcgrate.com /about/npcomprb/hcp (688 words)

	Week 1 Abstracts
		If the cycle is also a hamiltonian cycle, then G is said to be 1t k-ordered hamiltonian.
		A 2-factor is a 2-regular spanning subgraph of a graph, that is, the union of vertex dis joint cycles that spans the vertex set of the graph.
		Another shows that there is a long cycle by finding a collection of cycles and lower bound on the average length of the cycles in the collection.Yet another way to show that there are abundant long cycles is to show that they generate the cycle space.
	dimacs.rutgers.edu /drei/1998/week1.html (2432 words)

	The Hamiltonian Cycle Problem. (Site not responding. Last check: )
		A Hamiltonian cycle c of G is a cycle that goes through every vertex exactly once.
		Polymerase chain reaction (PCR) is a standard technique in molecular biology that cycles about 30 times through two or three temperature changes, uses DNA polymerase enzyme and produces an exponential increase of a given target molecule.
		The blue strand indicated with a dotted line is a circular single-stranded molecule that encodes the Hamiltonian Cycle.
	www.math.usf.edu /~jonoska/bio-comp/node3.html (893 words)

	Puzzle 359. First N primes in a circle
		The independent set rule show that G(N) is surely non Hamiltonian for 35 < N < 75, for 203 < N < 767 with possibly exception N = 759, and with N > 1341, at least up to 1000000 (i.e.
		Edwin Clark observed that for N >= 298 there is no Hamiltonian cycle with diagonally opposite vertices connected in G(N), as the 298 prime is 1973 connected only to 2 and 5.
		Hamiltonian cycle, we leave it as an exercise for the interested reader to
	www.primepuzzles.net /puzzles/puzz_359.htm (1487 words)

	PlanetMath: Hamiltonian cycle
		If there is a cycle visiting all vertices exactly once, we say that the cycle is a Hamiltonian cycle.
		See Also: Hamiltonian graph, Hamiltonian path, graph theory, traceable
		This is version 5 of Hamiltonian cycle, born on 2001-10-24, modified 2006-10-31.
	planetmath.org /encyclopedia/HamiltonianCycle.html (57 words)

	Jeremy Frank - Hamiltonian Cycle Research Page (Site not responding. Last check: )
		The Hamiltonian Cycle Problem is to decide whether or not there is a Hamiltonian Cycle in a given graph.
		We examined the phase transition for Hamiltonian Cycle and gave evidence supporting the theoretical result; our results suggest that the constant is smaller than Posa used in his result.
		This work is currently being extended to include a more detailed analysis of the computational costs of backtracking search for Hamiltonian Cycle as well as an investigation into the number of Hamiltonian Cycles found at the phase transition.
	ic-www.arc.nasa.gov /people/frank/ham.html (172 words)

	BackgroundMaterial (Site not responding. Last check: )
		Let C be any simple cycle in G. Remove the edges in C and find eulerian cycles in each component of the remaining subgraphs.
		Unlike the situation with eulerian graphs, it is apparently very difficult to test whether a graph is hamiltonian.
		Since G is connected, C and P is maximum, C must be a hamiltonian cycle.
	www.math.gatech.edu /~trotter/Section4-EulerHam.htm (435 words)

	IP2 A Second Hamiltonian Cycle (Site not responding. Last check: )
		The problem of finding a Hamiltonian cycle in a graph is NP-complete.
		In this presentation, the speaker will present a general sufficient condition for a second Hamiltonian cycle.
		Moreover, in a bipartite Hamiltonian graph, the number of Hamiltonian cycles increases (at least) exponentially as a function of the minimum degree.
	www.siam.org /meetings/da99/ip2.htm (177 words)

	Hamiltonian path - Wikipedia, the free encyclopedia
		A Hamiltonian path or traceable path is a path that visits each vertex exactly once.
		A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.
		Ore, O "A Note on Hamiltonian Circuits." American Mathematical Monthly 67, 55, 1960.
	en.wikipedia.org /wiki/Hamiltonian_path (659 words)

	[No title]
		Recall: A Hamiltonian cycle in a graph exists if there is a path that reaches every node and returns to the starting point as long as no node (except the first) is reached more than once and no edge is used more than once.
		Reduction#1: (problem 13.10) show that the Hamiltonian cycle problem for undirected graphs is reducible to the Hamiltonian cycle problem for directed graphs.
		If the answer is no, then there is no cycle that can reach every node and return using only n edges, and so no Hamiltonian cycle exists.
	www.nku.edu /~foxr/CSC464/PROBLEMS/prob13-reductions.doc (352 words)

	Hamiltonian Cycle (Site not responding. Last check: )
		A ``Hamiltonian cycle'' on G is a sequence of vertices (
		} is an edge of G. The problem is: write a program that, given a dense undirected graph G = (V; E) as input, determines whether G admits a Hamiltonian cycle on G and outputs that cycle, if there is one, or outputs ``
		The output file must contain the sequence of vertices that form a Hamiltonian cycle in the form:
	acm.uva.es /p/v7/775.html (167 words)

	4.3 Homework
		Give an example of a graph that has a Euler cycle, but not a Hamiltonian cycle.
		Give an example of a graph that has a Hamiltonian cycle but not an Euler cycle.
		If there is a Hamiltonian cycle, there will be exactly 12V and 12E.
	www.omegamath.com /Discrete/cp4.3.html (220 words)

	leapers
		A Hamiltonian cycle is a loopless, connected graph where every point of the graph has degree 2.
		A Hamiltonian path is a connected graph of N points with N-2 points of degree 2 and 2 points of degree 1.
		A Re-entrant knights tour is a Hamiltonian cycle on an 8x8 lattice where each line has length.
	www.mathpuzzle.com /leapers.htm (1909 words)

	Hamiltonian Cycle / Gray Code Animation (Site not responding. Last check: )
		A Hamiltonian Cycle is a cycle that visits each node exactly once.
		Hamiltonian cycles on hypercubes provide constructions for Gray codes, namely orderings of all subsets of n items such that neighboring subsets differ in exactly one element.
		Hamilitonian cycle is an NP-complete problem, so no worst-case efficient algorithm exists to find such a cycle.
	www.cs.sunysb.edu /~skiena/combinatorica/animations/ham.html (150 words)

	Hamiltonian cycle (Site not responding. Last check: )
		A Hamiltonian cycle includes each vertex once; an Euler cycle includes each edge once.
		Links to papers on many aspects of Hamiltonian cycles and paths.
		Paul E. Black, "Hamiltonian cycle", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology.
	www.nist.gov /dads/HTML/hamiltonianCycle.html (126 words)

	Graph Theory Lecture Notes 12
		Each vertex of the dodecahedron was labeled with the name of a city, and one was to find a circuit using the edges of the dodecahedron which visited each city once and only once.
		Corollary: G is Hamiltonian iff c(G) is Hamiltonian.
		There are corresponding theorems for the existence of a Hamiltonian cycle in digraphs.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtln12.html (723 words)

	Lecture 25: Hamiltonian Cycle Problem by COMS 482
		Proof: Hamiltonian Cycle is in NP because a given list of vertices can be checked for a tour in polynomial time.
		Walking left to right down a row means “true,” while right to left means “false.” In this way, a Hamiltonian cycle through the graph represents an assignment of truth values to the variables in a list of 3-SAT clauses.
		This entry was posted on Monday, March 28th, 2005 at 3:40 pm and is tagged with hamiltonian cycle problem, traveling salesman problem, polynomial time, truth assignment, true term, truth values, cycle c, side trip, 2n, clauses, vertex, graph, choose one, node, diversion, variables, np, lt.
	cs482.elliottback.com /2005/03/28/lecture-25-hamiltonian-cycle-problem (622 words)

	The Hamiltonian Page
		Parallel algorithms for the Hamiltonian cycle and Hamiltonian problems in semi-complete bipartite digraphs, Jorgen Bang-Jensen, Mohamed El Haddad, Yannis Manoussakis, and Teresa M.Przytycka.
		Optimal Parallel Construction of Hamiltonian Cycles and Spanning Trees in Random Graphs, by Philip D. MacKenzie and Quentin F. Stout, (preliminary version), In Proc.
		Phase Transitions in the Properties of Random Graphs, by J. Frank and C.U. Martel, Aug. 25, 1995.
	www.densis.fee.unicamp.br /~moscato/Hamilton.html (1318 words)

	Finding Hamiltonian Cycles: Algorithms, Graphs and Performance (Site not responding. Last check: )
		The problem consists of finding a tour (or cycle) that visits all the vertices once and returns to the starting vertex.
		Generalizations of the knight's tour problem (a subset of the Hamiltonian cycle problem).
		My goal here is to identify graphs and graph properties that make the Hamiltonian cycle problem hard or that distinguish between various Hamiltonian cycle algorithms.
	web.cs.ualberta.ca /~joe/Theses/vandegriend.html (90 words)

	Traveling Salesman Problem
		Please note in particular: except for the Hamiltonian cycle problems, all problems instances are defined on a complete graph and, at present, all distances are defined to be
		Given a set of n nodes and distances for each pair of nodes, find a Hamiltonian path from node 1 to node n of minimal length which takes given precedence constraints into account.
		Each precedence constraint requires that some node i has to be visited before some other node j.
	www.iwr.uni-heidelberg.de /groups/comopt/software/TSPLIB95 (437 words)

	[No title] (Site not responding. Last check: )
		A set of (directed) edges H (a subset of E) is a Hamiltonian cycle of G, if it is a cycle and goes through each vertex of G exactly once.
		OUTPUT: a set of edges of the graph forming its Hamiltonian cycle (or a message that the graph has no Hamiltonian cycles).
		In each case, the vertex set is {1,2,...,200}, the edge set has 1250 elements; only the edges are listed in the data sets given below.
	www.cs.uky.edu /ai/benchmark-suite/hamiltonian-cycle.html (104 words)

	[No title]
		Recall that we mentioned this type of cycle when we were examining Euler paths and Euler circuits.
		A graph is called a Hamiltonian graph if it includes at least one Hamiltonian cycle.
		There is no formula which characterizes a Hamiltonian graph as there was with an Euler graph.
	www.cs.ucf.edu /courses/cop3503h/day26.doc (249 words)

	Hamiltonian Cycles
		A Hamiltonian cycle on directed graph is a cyclic path that passes through every vertex exactly once, before returning to the start vertex.
		Given a directed graph G = (V,E), find a Hamiltonian cycle, if there is one.
		The cycle in the example would appear as follows.
	jupiter.clarion.edu /~rsmaby/cis356/hamilton.html (745 words)

	An Algorithmic Construction of Fault Free Hamiltonian Cycles in Faulty Hypercubes.
		A simple algorithm is presented which specifies a Hamiltonian cycle avoiding any set of n-2 edges in an n-dimensional hypercube.
		In the language of graph embedding, constructing a Hamiltonian circuit in a graph is the same a finding an optimal dilation-1 cycle embedding in that graph.
		Since a reflected Gray code Hamiltonian path has the property that the end vertices are adjacent, adding the edge between them gives a reflected Gray code Hamiltonian circuit.
	www.colostate.edu /dept/Mathematics/profiles/ochs/paper.html (1613 words)

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