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Topic: Petersen graph

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	Petersen (Site not responding. Last check: 2007-10-21)
		The Petersen graph is the complement of the Johnson graph J(5,2).
		The Petersen graph is also a cage (graph with smallest possible number of vertices given its valency and girth).
		The Petersen graph is contained in the complement of the Clebsch graph and the Sp(4,2) Generalized Quadrangle and the Hoffman-Singleton graph.
	www.win.tue.nl /~aeb/drg/graphs/Petersen.html (346 words)

	Encyclopedia: Graph theory (Site not responding. Last check: 2007-10-21)
		Graphs are usually represented pictorially using dots to represent vertexes, and arcs representing the edges between connected vertexes.
		In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the clique number of that subgraph.
		In the mathematical discipline of graph theory a covering for a graph is a set of vertices (or edges) so that the elements of the set are close (adjacent) to all edges (or vertices) of the graph.
	www.nationmaster.com /encyclopedia/Graph-theory (3291 words)

	Petersen graph -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		The Petersen graph is a small (A drawing illustrating the relations between certain quantities plotted with reference to a set of axes) graph that serves as a useful example and counterexample in (Click link for more info and facts about graph theory) graph theory.
		The Petersen graph frequently arises as a counterexample or exception in graph theory.
		The Petersen graph family consists of the seven graphs that can be formed from the (Click link for more info and facts about complete graph) complete graph by zero or more applications of delta-Y or Y-delta transforms.
	www.absoluteastronomy.com /encyclopedia/p/pe/petersen_graph.htm (668 words)

	Encyclopedia: Petersen Graph (Site not responding. Last check: 2007-10-21)
		In graph theory, similar to its vertex counterpart, an edge coloring of a graph, when used without any qualification, is always assumed to be a proper coloring on the edges, meaning no two incident edges are assigned the same color.
		In graph theory the complement or inverse of a graph is a graph on the same vertices such that two vertices of are adjacent if and only if they are not adjacent in.
		In graph theory, the line graph L(G) of a graph G is a graph such that each vertex of L(G) represents an edge of G; and any two vertices of L(G) are adjacent if and only if their corresponding edges are incident, meaning they share a common...
	www.nationmaster.com /encyclopedia/Petersen-Graph (1384 words)

	Logo (Site not responding. Last check: 2007-10-21)
		The particular graph we are using is known as the Petersen graph after the 19th century Danish mathematician, Julius Petersen.
		Assigning colours to the vertices of a graph in such a way that no two vertices that are joined by an edge have the same colour is known as a proper colouring of the vertices.
		Assigning colours to the edges of a graph in such a way that no two edges that meet at a vertex have the same colour is known as a proper colouring of the edges.
	mathcentral.uregina.ca /BB/logo.html (290 words)

	PlanetMath: Petersen graph (Site not responding. Last check: 2007-10-21)
		An example of graph that is traceable but not Hamiltonian.
		This is also the canonical example of a hypohamiltonian graph.
		This is version 5 of Petersen graph, born on 2001-10-24, modified 2002-02-21.
	planetmath.org /encyclopedia/PetersensGraph.html (72 words)

	PlanetMath: Moore graph (Site not responding. Last check: 2007-10-21)
		The automorphism group of the pentagon is the dihedral group with 10 elements.
		And the one of the HS graph is isomorphic to
		This is version 4 of Moore graph, born on 2005-04-11, modified 2005-04-15.
	planetmath.org /encyclopedia/MooreGraph.html (326 words)

	A nice puzzle modeled on the Petersen graph
		A Hamiltonian circuit on the Petersen graph, if existed, would contain 4 edges of at least one of the three groups: external, middle, or internal.
		If the Petersen graph has a Hamiltonian circuit, then the edges of the Petersen graph can be colored in three colors (the edges are 3-colored if the edges are colored in 3 colors and no two edges with a common node has the same color).
		Hence, to show that the Petersen graph has no Hamiltonian circuit, it suffices to show that the edges of the Petersen graph cannot be 3-colored.
	www.cut-the-knot.com /pigeonhole/Petersen.shtml (686 words)

	Solving the Petersen Graph Zome Challenge
		To solve the Petersen Graph Zome Challenge, I wrote a computer program that methodically enumerates all Zome constructions that are topologically equivalent to the Petersen Graph.
		For each Petersen Graph found, the program prints the three cycles using the nomenclature described in Analytic Zome, the number of intersections, and the length of the 2-to-9 segment (based on the other 14 segments having length 1).
		Relaxing the restriction on the length of the 15th strut allows for many non-intersecting Petersen graphs that are mostly Zome construcible (the length of the 15th strut is often not a Zome length) and even some that are completely Zome constructible (the length of the 15th strut is a Zome length).
	www.rawbw.com /~davidm/zome/solvingPetersen.html (534 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 (8629 words)

	Petersen graph
		The Petersen graph is a small graph that serves as a useful example and counterexample in graph theory.
		The Petersen graph is the smallest cubic graph[?] that has no Hamiltonian cycle, the smallest cubic graph of girth 5, and the largest cubic graph with diameter 2.
		It is named for the Danish mathematician Julius Petersen.
	www.ebroadcast.com.au /lookup/encyclopedia/pe/Petersen_graph.html (97 words)

	Hoffman-Singleton graph
		As we saw, Γ is subgraph of the Higman-Sims graph on 100 vertices.
		The graph on the 15-cocliques, adjacent when they meet in 0 or 8 is the Higman-Sims graph.
		The subgraph induced on the orbit of size 30 is Tutte's 8-cage, the incidence graph of the generalized quadrangle GQ(2,2).
	www.win.tue.nl /~aeb/drg/graphs/Hoffman-Singleton.html (914 words)

	Week 3 Abstracts
		The random bipartite model G(n,n,p) is a probability space, whose elements are labeled bipartite graphs with vertex classes A and B, both of size n, where a vertex from A and that from B are connected by an edge randomly and independently with probability p=p(n).
		We prove that every graph on the Klein Bottle which does not contain contractible cycles of length 3 or 4 is either 3-colorable or has a subgraph isomorphic to a member of a particular family of non-3-colorable graphs.
		The Kneser graph K(n,k) is the graph of the disjointness relation on the k-element subsets of an n-set.
	dimacs.rutgers.edu /drei/1998/week3.html (3950 words)

	Graph Theory Glossary - h (Site not responding. Last check: 2007-10-21)
		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)

	ipedia.com: Snark (graph theory) Article (Site not responding. Last check: 2007-10-21)
		In graph theory, a snark is a bridgeless cubic graphss with edge chromatic number of at least four.
		In other words, it is a graph in which every node has three branches, and the nodes cannot be coloured in fewer than four colours without two nodes of the same colour meeting at a point.
		These graphs are part of the proof of the four color theorem.
	www.ipedia.com /snark__graph_theory_.html (178 words)

	Planar Graphs
		A plane graph is one that has been drawn in the plane in such a way that its edges intersect only at their common end-vertices.
		Two graphs, G and H are defined to be homeomorphic if you can make one graph into the other by inserting vertices of degree 2.
		Let G be a plane graph, and consider the regions bounded by the edges of G.
	www.math.lsa.umich.edu /mmss/coursesONLINE/graph/graph5 (734 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)

	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)

	The Very Best Books : The Petersen Graph (Australian Mathematical Society Lecture Series)
		The Petersen graph is a small graph which shows up amazingly often as a counterexample in graph theory.
		Graph theory is kind of a hobby of mine, and I find this book an absorbing mishmash of introductory material.
		A little knowledge of graphs would be good, and you should know what groups are, and be able to read a proof.
	www.elise.com /store/Reviews/ItemId/0521435943 (170 words)

	flows (Site not responding. Last check: 2007-10-21)
		Suppose that a graph is embedded in an orientable surface such that the faces are properly 5-colored.
		The value 5 cannot be strengthened since the Petersen graph does not have a nowhere-zero 4-flow.
		A weaker conjecture was that every cubic bridgeless graph without a subdivision of the Petersen graph is 3-edge-colorable.
	www.emba.uvm.edu /~archdeac/problems/flows.htm (352 words)

	[No title]
		Determine the 6-cage.ªh$ ó>8¨1The Petersen Graph and its Generalizations G(n,k)ª1$ ¨»Petersen graph G(5,2) is an example of a generalized Petersen graph G(n,k).
		One may argue that the main topic of graph theory is the study of graph invariants.
		Graph invariant is a property, (usually a number), that is preserved under an isomorphism.¡À-ZZZ Z[Z 7LªÓ$ ó¨Isomorphism - Exercisesª$ ¨lN1.
	www.fmf.uni-lj.si /~pisanski/Predavanja/Konfiguracije/Chapter1_03_graphexamples.ppt (541 words)

	Ed Pegg's Math Games - Domino Graphs
		In alternate terminology, we conclude that the Petersen Graph is non-Hamiltonian.
		For a bigger graph, consider the Stomachion of Archimedes, which Bill Cutler recently solved (my 11-17-03 column).
		As a graph, we can connect any two solutions that related by one of these flips or rotations.
	www.maa.org /editorial/mathgames/mathgames_12_15_03.html (724 words)

	Math Games: Cubic Symmetric Graphs
		Foster Census, which is a listing of all of the cubic symmetric graphs up to 768 vertices.
		The Petersen graph can be drawn with just 2 crossings, and the Coxeter graph with just 11.
		In a zero-symmetric graph, the vertices are equivalent, but the edges are maximally dissimilar.
	www.maa.org /editorial/mathgames/mathgames_12_29_03.html (982 words)

	Petersen (Site not responding. Last check: 2007-10-21)
		Julius Petersen wrote a series of school and undergraduate texts which achieved international acclaim despite being too difficult for all but the ablest pupils.
		A paper which he wrote in 1891 marks the birth of the theory of regular graphs.
		the work of Petersen and Zeuthen is regarded as being responsible for the emergence of Danish mathematics on the international scene towards the end of the 19th century.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Petersen.html (131 words)

	Graph Theory Lesson 12 (Site not responding. Last check: 2007-10-21)
		An Euler circuit on a graph G is a circuit that visits each vertex of G and uses every edge of G.
		An Euler path on a graph G is a path that visits each vertex of G and uses every edge of G.
		A graph that has directed edges is called a directed graph or sometimes just a digraph.
	www.utc.edu /Faculty/Christopher-Mawata/petersen/lesson12.htm (469 words)

	Untitled Document
		For awhile, only one cubic graph was known that could not be Tait-colored: the famous Petersen graph.
		Graphs which could not be Tait-colored were called Snarks by Martin Gardner, since like the Lewis Carroll beastie, they were hard to find.
		Petersen-free means that a Petersen graph cannot be reached by deleting and/or contracting edges (a minor).
	www.mathpuzzle.com /4Dec2001.htm (6140 words)

	On the Relation between the Petersen Graph and the Characteristic of Separating Cuts of Matching Covered Graphs ... (Site not responding. Last check: 2007-10-21)
		Abstract: A matching covered graph is a connected graph each edge of which lies in some perfect matching.
		A cut of a matching covered graph is separating if each of its two contractions yields a matching covered graph.
		The characteristic of a nontight separating cut is the smallest number of edges greater than one that some perfect matching of the graph...
	citeseer.ist.psu.edu /campos00relation.html (297 words)

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