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Topic: Tournament (graph theory)

	PlanetMath: graph theory
		Graph theory is the branch of mathematics that concerns itself with graphs.
		Now, a (finite) graph is usually thought of as a subset of pairs of elements of a finite set (called vertices), or more generally as a family of arbitrary sets in the case of hypergraphs.
		This is version 24 of graph theory, born on 2002-10-22, modified 2006-09-19.
	planetmath.org /encyclopedia/GraphTheory.html (505 words)

	PlanetMath: tournament
		A tournament is a directed graph obtained by choosing a direction for each edge in an undirected complete graph.
		The name “tournament” originates from such a graph's interpretation as the outcome of some sports competition in which every player encounters every other player exactly once, and in which no draws occur; let us say that an arrow points from the winner to the loser.
		This is version 3 of tournament, born on 2002-10-11, modified 2003-05-17.
	planetmath.org /encyclopedia/Tournament.html (182 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)

	05C: Graph theory
		A graph is a set V of vertices and a set E of edges -- pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets, which is clearly a broad topic.
		A graph may be viewed as a one-dimensional CW-complex and hence studied with tools from Algebraic Topology, in particular, questions of planarity (and genus).
		Determining the genus of a graph is NP-complete.
	www.math.niu.edu /~rusin/known-math/index/05CXX.html (1204 words)

	Application to Graph theory
		Graph Theory is now a major tool in mathematical research, electrical engineering, computer programming and networking, business administration, sociology, economics, marketing, and communications; the list can go on and on.
		In the above graph, the vertices A, C and E have the following property: from each one there is either a 1-step or a 2-step connection to any other vertex in the graph.
		In any dominance-directed graph there is at least one vertex from which there is a 1-step or a 2-step connection to any other vertex in the graph.
	aix1.uottawa.ca /~jkhoury/graph.htm (1110 words)

	Graphs, Euler and Hamilton circuits
		The subject we now call graph theory, and perhaps the wider topic of topology, was founded on the work of Leonhard Euler, and a single famous problem called the Seven Bridges problem. Probably the first paper on graph theory was by Euler.
		Here is a simple example, a quadrilateral drawn with edges connecting every possible pair of points. Such a graph is called a complete 4-graph and is one of a group of similar graphs of n vertices called compete n-graphs.
		A special type of complete directed graph that has a single directed graph between each pair of vertices is called a tournament graph or a complete oriented graph.
	www.pballew.net /graphs.html (840 words)

	Methodology
		Graph transformation systems, or graph grammars, are a branch of graph theory research that rigorously defines mathematical operations such as addition and intersection in graphs.
		When viewing the artifact as a graph constructed from an initial simpler graph that describes the problem, one needs to develop a set of rules to capture the valid transformations that can occur.
		In genetic algorithms, this is the “survival of the fittest” tournament selection where candidates with inferior fitness values are removed from the search process.
	www.me.utexas.edu /~adl/graphsynth/introMethod.htm (893 words)

	Discrete Mathematics Project -- Checker Tournament (Site not responding. Last check: 2007-10-10)
		This activity focuses on applying graph theory to derive a schedule to meet the criteria of a tournament.
		This activity could be used to introduce graph theory such as to develop a schedule.
		Conduct an actual Checker Tournament in the classroom with each group as one player and discuss their schedules, which was the easiest to follow, which schedule was more time efficient, etc.
	www.colorado.edu /education/DMP/activities/graph/amsact03.html (424 words)

	Graph Theory
		Graph Theory was born to study problems of this type.
		In an undirected graph, this is obviously a metric.
		Bound δ (of a graph embedded in on a surface)
	www.math.fau.edu /locke/GRAPHTHE.HTM (1165 words)

	Info on Score Sequences
		A tournament is a complete graph in which every edge has a orientation.
		A score sequence of a tournament is a monotonically non-decreasing sequence of the out-degrees of its vertices.
		Score sequences do not characterize tournaments; there are non-isomorphic tournaments with the same score sequence.
	www.theory.csc.uvic.ca /~cos/inf/nump/ScoreSequence.html (245 words)

	Tournament (graph theory) - Wikipedia, the free encyclopedia
		Any tournament on a finite number n of vertices contains a Hamiltonian path, i.e., directed path on all n vertices.
		This is easily shown by induction on n: suppose that the statement holds for n, and consider any tournament T on n + 1 vertices.
		As a corollary, a tournament is strongly connected if and only if it has a Hamiltonian cycle.
	en.wikipedia.org /wiki/Tournament_(graph_theory) (281 words)

	"Introduction to Graph Theory - new problems"
		Determine whether the graph obtained by deleting a diagonal edge is isomorphic to the graph obtained by deleting one of the edges on the cycle.
		Count the spanning trees in a graph that is the union of a k-cycle and an l-cycle with one common edge.
		(!) The Kneser graph K(n,k) is the disjointness graph of the k-element subsets of [n].
	www.math.uiuc.edu /~west/igt/newprob.html (10012 words)

	Amazon.com: Handbook of Graph Theory: Books: Jonathan L. Gross,Jay Yellen (Site not responding. Last check: 2007-10-10)
		The Handbook of Graph Theory is the most comprehensive single-source guide to graph theory ever published.
		Since the main application of graph theory is in computer science, much of the Handbook relates to that field, yet there is no chapter on computational complexity.
		The interest of this one is to have pointers to the literature and to cover as much as possible of graph theory, so don't expect to find much details about something particular.
	www.amazon.com /Handbook-Graph-Theory-Jonathan-Gross/dp/1584880902 (1213 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.
		If there is more than one Hamiltonian path in a tournament, the vertices do not have a unique ranking.
		This is a contradiction since D is a tournament.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtln12.html (723 words)

	The Graph Theorists' Home Page Guide
		First of all, if you're a graph theorist or some person with strong interest in graph theory (you need not to be a mathematician!), and if you have a homepage but don't find a link to it on this page, please contact me as described above.
		PIGALE is a graph editor with an interface to the LEDA library and with many algorithms implemented essentially concerning planar graphs.
		"Graph Theory and Its Applications" (together with Jay Yellen), "a comprehensive applications-driven textbook that provides material for several different courses in graph theory." This site also provides links to other graph theoretical and mathematical resources.
	www.joergzuther.de /math/graph/homes.html (8736 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Cauchy's integral was successful in its goal of integrating continuous functions, but the budding theory of Fourier series prompted the need to integrate a much larger class of functions.
		While writing his 1934 Ergebnisse monograph, which was an attempt to describe the ideas of the theory of surfaces developed by the Italian geometers, Zariski "became convinced that the whole structure must be done again by purely algebraic methods." Thus began Zariski's remarkable contributions to algebraic geometry.
		Graph theory and statistics are used to extend the concept of the big loser to double, triple, and n-elimination tournaments, and also to attempt to find out what seed is most likely to become the Big Loser.
	home.earthlink.net /~kymaa/meetings/abstracts03.doc (1628 words)

	Combinatorics and Graph Theory (Site not responding. Last check: 2007-10-10)
		Faculty in the research area of Combinatorics and Graph Theory are:
		He is a leading researcher in topological graph theory and is managing editor of the highly respected Journal of Graph Theory.
		Her principal area of research currently is real number graph labeling with distance conditions, which has been recently developed from original integer graph labeling and generalized graph coloring, with applications in radio channel assignment of mobile networks.
	www.emba.uvm.edu /math/research/combo.php (231 words)

	Glossary of graph theory - Wikipedia, the free encyclopedia
		An independent set, or stable set or staset, is a set of isolated vertices, i.e., no pair of vertices is adjacent.
		A digraph, or directed graph, or oriented graph, is analogous to an undirected graph except that it contains only arcs.
		A mixed graph may contain both directed and undirected edges; it generalizes both directed and undirected graphs.
	en.wikipedia.org /wiki/Glossary_of_graph_theory (5943 words)

	List of Seminar & Colloquia held at UMD Math. and Stat.
		Graph theory is a very useful tool for scheduling round robin tournaments.
		Optimal foraging theory consists of a collection of models that represent how animals choose and look for food.
		Bayesian foraging models are used to study how animals use information about the environment to decide when to leave one food patch and move on to another.
	www.d.umn.edu /math/seminar (1007 words)

	Tournament Organization Resources Sports
		- Provider of tournament registration and management services for sports in the US and Canada.
		- Advice for orgainzers of Go tournaments, with particular emphasis on the judicial aspects (which would apply equally to a tournament of any similar game).
		Schedules for such all-play-all tournaments up to 26 players, and for carry-over tournaments up to 16+16=32 players, in HTML, PDF and txt formats.
	www.iaswww.com /ODP/Sports/Resources/Tournament_Organization (146 words)

	Math Games: Tournament Dice (Site not responding. Last check: 2007-10-10)
		The graph seen above is a directed graph -- it has arrows on each edge between letters.
		When a digraph is complete - with an arrow between any pair of points, it's called a Tournament graph.
		A tournament graph for this is below, which fl/grey lines instead of arrows.
	www.maa.org /editorial/mathgames/mathgames_07_11_05.html (816 words)

	18.310 - Fall 2005
		I went over the algorithms in a different order, and omitted tournament sort from my lecture.
		For the Wednesday 19th, lecture, I was planning on just doing the basic definitions from graph theory, and Kuratowski's theorem from the lecture notes.
		While preparing for the class, I realized that not only is the proof of Kuratowski's theorem in the OCW lecture notes wrong, but I think it's wrong in more than one way.
	www-math.mit.edu /~shor/PAM/18.310.html (1154 words)

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