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Topic: Hamiltonian graph

Related Topics

Weighted graph

Hamiltonian cycle

Circle graph

Hamiltonian path

Complete graph

	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_path (639 words)

	Graphs Glossary
		A graph is bipartite if the vertices can be partitioned into two sets, X and Y, so that the only edges of the graph are between the vertices in X and the vertices in Y. Trees are examples of bipartite graphs.
		A chain in a graph is a sequence of vertices from one vertex to another using the edges.
		An induced (generated) subgraph is a subset of the vertices of the graph together with all the edges of the graph between the vertices of this subset.
	www-math.cudenver.edu /~wcherowi/courses/m4408/glossary.htm (1926 words)

	The Hamiltonian Page
		A sufficient condition for a semicomplete multipartite digraph to be Hamiltonian, by J. Bang-Jensen, G. Gutin and J. Huang.
		Sufficient conditions for semicomplete multipartite digraphs to be Hamiltonian, by Y. Guo, M. Tewes, L. Volkmann, 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)

	Graph Theory Glossary - h (Site not responding. Last check: 2007-10-10)
		Graph obtained by successfully adding edges between vertices whose degree-sum is as large as the number of vertices.
		A connected graph consisting of two vertex-disjoint polygons and a minimal (not necessarily minimum-length) connecting path (this is a loose handcuff), or of two polygons that meet at a single vertex (a tight handcuff or figure eight).
		A connected labeled graph with n edges in which all vertices can be labeled with distinct integers (mod n) so that the sums of the pairs of numbers at the ends of each edge are also distinct (mod n).
	www.cc.ioc.ee /jus/gtglossary/gtglos_h.htm (2413 words)

	Glossary of graph theory - Wikipedia, the free encyclopedia
		Graph theory is a growth area in mathematical research, and has a large specialized vocabulary.
		Likewise, a graph G is said to be homomorphic to a graph H if there is a mapping, called a homomorphism, from V(G) to V(H) such that if two vertices are adjacent in G then their corresponding vertices are adjacent in H.
		A vertex of degree 0 is an isolated vertex.
	en.wikipedia.org /wiki/Glossary_of_graph_theory (5943 words)

	Graph Theory
		Example: This graph is not simple because it has 2 edges between the vertices A and B. Two vertices, v and w, of a graph are ADJACENT if there is an edge, vx, joining theem; the vertices are then considered INCIDENT to the edge, vx.
		In terms of graph theory, in any graph the sum of all the vertex-degrees is an even number - in fact, twice the number of edges.
		Like the graph 1 above, if a graph has a path that includes every vertex exactly once, while ending at the initial vertex, the graph is Hamiltonian (is a Hamiltonian graph).
	jwilson.coe.uga.edu /EMAT6680/Yamaguchi/emat6690/essay1/GT.html (1210 words)

	Some definitions in Graph Theory
		In a directed graph G, a flow is the assignment of a real value f(e) to each arc e of G such that, for each vertex, the sum of entering flows is equal to the sum of leaving flows (Kirchhoff's law).
		A Hamiltonian cycle of a graph G is an elementary cycle going through all vertices of G. A graph is Hamiltonian if it has a Hamiltonian cycle.
		A graph is called Eulerian if such a closed walk exists; this is equivalent to the graph being connected and with even degrees for all vertices (this is a theorem of Euler, frequently considered the first theorem in Graph Theory).
	www-leibniz.imag.fr /GRAPH/english/definitions.html (1017 words)

	Graph Theory Open Problems
		A graph which can be embedded in the plane so that vertices correspond to points in the plane and edges correspond to unit-length line segments is called a ``unit-distance graph.'' The question above is equivalent to asking what the chromatic number of unit-distance graphs can be.
		It is known that this is not true if you remove the "bipartite" condition, but the smallest known such graph which is not Hamiltonian has 38 vertices, as shown to the right.
		To get the square of an oriented graph (or any directed graph) you leave the vertex set the same, keep all the arcs, and for each pair of arcs of the form (u,v), (v,w), you add the arc (u,w) if that arc was not already present.
	dimacs.rutgers.edu /~hochberg/undopen/graphtheory/graphtheory.html (705 words)

	Finding a Hamiltonian Circuit
		In a digraph, a hamiltonian circuit is a path that travels through every vertex once, and winds up where it started.
		When the entire graph is constructed, the circuit will start from the right if x is true, and from the left if x is false.
		Let h be a hamiltonian graph, with a hamiltonian circuit.
	www.mathreference.com /lan-cx-np,hamil.html (888 words)

	Algorithm for Hamiltonian Circuits - Ashay Dharwadker
		Graph of the octahedron with a Hamiltonian circuit
		Graph of the icosahedron with a Hamiltonian circuit
		Thus, a re-entrant knight's tour on the chessboard corresponds to a Hamiltonian circuit in the knight's graph.
	www.geocities.com /dharwadker/hamilton/main.html (4446 words)

	Graph Theory (math 224)
		A plane graph is a graph which is actually embedded in the plane so that each vertex corresponds to a point and each edge to a simple closed curve (or straight-line segment if you prefer) joining the points corresponding to its endpoints.
		The complement of a plane graph is a disjoint union of connected components which are called the _regions_ of the plane graph.
		Similarly, for graphs in the _torus_ (think "doughnut" or "inner tube") n-m+r = 0 and the corresponding upper bound on edges is m leq 3n; hence, average degree is at most 6 and so there must be a vertex of degree not exceeding 6 in any toroidal graph.
	www.georgetown.edu /faculty/kainen/graph-theory.html (3496 words)

	Dave Witte Morris' papers in graph theory
		This paper finds all the hamiltonian paths in the cartesian product of two directed cycles, and uses these paths to prove there is a hamiltonian circuit in the cartesian product of any three or more directed cycles.
		This paper shows that any regular graph of infinite valence that has a two-way-infinite hamiltonian path either has infinite connectivity or can be constructed in a simple way by combining two graphs with infinite connectivity.
		This yields the first known construction of Cayley graphs for which 2g - g' is arbitrarily large, where g and g' are the orientable genus and the non-orientable genus of the Cayley graph.
	people.uleth.ca /~dave.morris/GraphTheory.shtml (1225 words)

	Week 1 Abstracts
		For example, we show that for a graph G on n vertices and any positive integer k, if the minimum degree of G is at least (rz + k - 3)/2 for k odd or at least (n + k-2)/2 for k even, then G is 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.
		For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices there is a cycle that encounters the vertices of the sequence in the given order.
	dimacs.rutgers.edu /drei/1998/week1.html (2432 words)

	CS 260-01 Chapter 7 Notes
		If a simple graph with n vertices (n>=3) such that for all distinct nonadjacent vertices i and j, the degree of i + degree of j exceeds n, then the graph is Hamiltonian-connected.
		It is a subgraph of a planar graph.
		A bipartate graph, by definition, is a graph between all the vertices of one set with all those of a second set; hence, there are only two sets or two colors.
	www.wpunj.edu /irt/courses/CS260rv/cs_notes7.htm (2004 words)

	PlanetMath: Hamiltonian graph
		A useful condition both necessary and sufficient for a graph to be Hamiltonian is not known.
		be a graph of order at least 3.
		This is version 8 of Hamiltonian graph, born on 2001-10-24, modified 2006-10-23.
	planetmath.org /encyclopedia/HamiltonianGraph.html (118 words)

	Graph Theory Lecture Notes 9a
		A hamiltonian graph is a graph that has a hamiltonian cycle.
		Determining whether or not a graph is hamiltonian is an NP-complete problem.
		The graph can always be assumed to be a complete graph by adding missing edges to the given graph and assigning them extremely large weights.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtaln9.html (537 words)

	The Story of Louis Pósa
		Also a graph need not contain every possible edge which could be drawn, that is, there are in general many different graphs with the same set of vertices.
		While a Hamiltonian circuit always provides a Hamiltonian path, upon the deletion of any edge, a Hamiltonian path may not lead to a Hamiltonian circuit (it depends on whether or not the first and last vertices of the path happen to be joined by an edge in the graph).
		Suppose the given graph G which has n vertices and every degree at least n/2 is non-Hamiltonian, that is, does not possess a Hamiltonian circuit.
	www.math.uwaterloo.ca /navigation/ideas/articles/honsberger/index.shtml (2708 words)

	Unsolved Problems
		An (m,n)-cage is an m-regular graph with girth n and, subject to this, with the least possible number of vertices.
		The bandwidth of a graph G is the minimum bandwidth among adjacency matrices of graphs isomorphic to G.
		Since you either attended the EIDMA Workshop on Hamiltonicity of 2-Tough Graphs or at least were invited to do so, we guess you are interested in the fact that we have refuted the conjecture that every 2-tough graph is hamiltonian.
	www.math.fau.edu /locke/Unsolved.htm (2911 words)

	Lecture 25: Hamiltonian Cycle Problem by COMS 482
		Given a graph G, is there “a path that visits each vertex once and return to the start,” i.e.
		Proof: Hamiltonian Cycle is in NP because a given list of vertices can be checked for a tour in polynomial time.
		This entry was posted on Monday, March 28th, 2005 at 3:40 pm and is tagged with hamiltonian cycle problem, graph, polynomial time, 3 sat, clauses, np complete, proof, true term, traveling salesman problem, variables, cycle c, truth assignment, boolean formula, truth values.
	cs482.elliottback.com /2005/03/28/lecture-25-hamiltonian-cycle-problem (764 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)

	The Hamiltonian Page (Site not responding. Last check: 2007-10-10)
		Hamiltonian Path Problems in the Online-Optimization of flexible manufacturing systems, by N. Ascheuer, Ph.D.Thesis, University of Technology Berlin, Germany, 1995.
		Groups & Graphs, by Bill Kocay, is a software package for directed and undirected graphs, combinatorial designs, and their automorphism groups.
		Optimal Parallel Construction of Hamiltonian Cycles and Spanning Trees in Random Graphs, by Philip D. MacKenzie and Quentin F. Stout, (preliminary version), In Proc.
	www.densis.fee.unicamp.br /~moscato/Hamilton.html (1318 words)

	distributed
		A Hamiltonian path exists in a graph only if there is a connected path from a designated starting node to a designated output node that visits each node of the graph exactly once.
		There are algorithms that can find a Hamiltonian path in an arbitrary graph, but their time complexity grows exponentially with the size of the graph.
		The graph is the one used by Adleman in Ref. 3, except for the extra edge between nodes 0 and 2.
	www.isi.edu /~lerman/etc/distributed.html (1740 words)

	Puzzle 359. First N primes in a circle
		As noted recently in Graphnet forum, a graph with at most 2n vertices is surely non Hamiltonian if it has an independent set of size at least n.
		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.
		Hamiltonian cycle, we leave it as an exercise for the interested reader to
	www.primepuzzles.net /puzzles/puzz_359.htm (1487 words)

	BackgroundMaterial (Site not responding. Last check: 2007-10-10)
		A connected graph G is eulerian if and only if every vertex has even degree.
		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)

	[No title]
		Figure 3 has more graphs like that (page 372) (draw them, you'll need them for example 3) Example 3: figure a has 8 edges and n = 5 vertices 1/2(n-1)(n-2)+2 = 8, so it satisfies thm 2, and yep, it is Hamiltonian.
		Consider the graph with 2^n vertices whose vertices are labelled with all the n bit strings of 0s and 1s.
		A graph is called a complete bipartite graph if, in addition, all every vertex of V1 is joined to every vertex of V2 by a unique edge.
	noether.uoregon.edu /~dolan/lecturenotes/notes10.html (1012 words)

	[No title]
		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.
		However, it should be obvious that all complete graphs are Hamiltonian.
	www.cs.ucf.edu /courses/cop3503h/day26.doc (249 words)

	Lecture 22 - techniques for proving hardness
		This new graph has a Traveling Salesman tour of cost n iff the graph is Hamiltonian.
		A vertex cover instance consists of a graph and a constant k, the minimum size of an acceptable cover.
		This has a Hamiltonian path from start to stop iff the original graph has a vertex cover of size k.
	www.cs.sunysb.edu /~algorith/lectures-good/node22.html (1403 words)

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