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| {{Infobox graph
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| | name = Petersen graph
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| | image = [[Image:Petersen1 tiny.svg|200px]]
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| | image_caption = The Petersen graph is most commonly drawn as a pentagon with a pentagram inside, with five spokes.
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| | namesake = [[Julius Petersen]]
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| | vertices = 10
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| | edges = 15
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| | automorphisms = 120 (S<sub>5</sub>)
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| | radius = 2
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| | diameter = 2
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| | girth = 5
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| | chromatic_number = 3
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| | chromatic_index = 4
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| | fractional_chromatic_index = 3
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| | properties = [[Cubic graph|Cubic]]<br/>[[Strongly regular graph|Strongly regular]]<br/>[[Distance-transitive graph|Distance-transitive]]<br/>[[Snark (graph theory)|Snark]]
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| }}
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| In the [[mathematics|mathematical]] field of [[graph theory]], the '''Petersen graph''' is an [[undirected graph]] with 10 [[vertex (graph theory)|vertices]] and 15 [[edge (graph theory)|edges]]. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named for [[Julius Petersen]], who in 1898 constructed it to be the smallest [[Bridge (graph theory)|bridgeless]] [[cubic graph]] with no three-edge-coloring.<ref>{{citation|url=http://www.win.tue.nl/~aeb/drg/graphs/Petersen.html|title=The Petersen graph|first=Andries E.|last=Brouwer|authorlink=Andries Brouwer}}</ref>
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| Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by {{harvs|first=A. B.|last=Kempe|authorlink=Alfred Kempe|year=1886|txt}}. Kempe observed that its vertices can represent the ten lines of the [[Desargues configuration]], and its edges represent pairs of lines that do not meet at one of the ten points of the configuration.
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| [[Donald Knuth]] states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general."<ref>{{citation|first=Donald E.|last=Knuth|title=[[The Art of Computer Programming]]; volume 4, pre-fascicle 0A. A draft of section 7: Introduction to combinatorial searching}}</ref>
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| == Constructions ==
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| [[File:Kneser graph KG(5,2).svg|thumb|left|Petersen graph as Kneser graph <math>KG_{5,2}</math>]]
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| The Petersen graph is the [[complement graph|complement]] of the [[line graph]] of <math>K_5</math>. It is also the [[Kneser graph]] <math>KG_{5,2}</math>; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an edge if and only if the corresponding 2-element subsets are disjoint from each other. As a Kneser graph of the form <math>KG_{2n-1,n-1}</math> it is an example of an [[odd graph]].
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| Geometrically, the Petersen graph is the graph formed by the vertices and edges of the [[hemi-dodecahedron]], that is, a [[dodecahedron]] with opposite points, lines and faces identified together.
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| == Embeddings ==
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| The Petersen graph is [[planar graph|nonplanar]]. Any nonplanar graph has as [[minor (graph theory)|minor]]s either the [[complete graph]] <math>K_5</math>, or the [[complete bipartite graph]] <math>K_{3,3}</math>, but the Petersen graph has both as minors. The <math>K_5</math> minor can be formed by contracting the edges of a [[perfect matching]], for instance the five short edges in the first picture. The <math>K_{3,3}</math> minor can be formed by deleting one vertex (for instance the central vertex of the 3-symmetric drawing) and contracting an edge incident to each neighbor of the deleted vertex.
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| [[Image:Petersen graph, two crossings.svg|thumb|right|The Petersen graph has [[Crossing number (graph theory)|crossing number]] 2.]]
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| The most common and symmetric plane drawing of the Petersen graph, as a pentagram within a pentagon, has five crossings. However, this is not the best drawing for minimizing crossings; there exists another drawing (shown in the figure) with only two crossings. Thus, the Petersen graph has crossing number 2. On a [[torus]] the Petersen graph can be drawn without edge crossings; it therefore has [[genus (mathematics)|orientable genus]] 1.
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| [[Image:Petersen graph, unit distance.svg|thumb|right|The Petersen graph is a [[unit distance graph]]: it can be drawn in the plane with each edge having unit length.]]
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| The Petersen graph can also be drawn (with crossings) in the plane in such a way that all the edges have equal length. That is, it is a [[unit distance graph]].
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| The simplest [[surface|non-orientable surface]] on which the Petersen graph can be embedded without crossings is the [[projective plane]]. This is the embedding given by the [[hemi-dodecahedron]] construction of the Petersen graph. The projective plane embedding can also be formed from the standard pentagonal drawing of the Petersen graph by placing a [[cross-cap]] within the five-point star at the center of the drawing, and routing the star edges through this cross-cap; the resulting drawing has six pentagonal faces. This construction forms a [[Regular map (graph theory)|regular map]] and shows that the Petersen graph has [[genus (mathematics)|non-orientable genus]] 1.
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| == Symmetries ==
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| The Petersen graph is [[strongly regular graph|strongly regular]] (with signature srg(10,3,0,1)). It is also [[symmetric graph|symmetric]], meaning that it is [[edge-transitive graph|edge transitive]] and [[vertex-transitive graph|vertex transitive]]. More strongly, it is 3-arc-transitive: every directed three-edge path in the Petersen graph can be transformed into every other such path by a symmetry of the graph.<ref>{{citation|first=László|last=Babai|authorlink=László Babai|contribution=Automorphism groups, isomorphism, reconstruction|id=Corollary 1.8|title=Handbook of Combinatorics|url=http://www.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps|pages=1447–1540|editor1-first=Ronald L.|editor1-last=Graham|editor1-link=Ronald Graham|editor2-first=Martin|editor2-last=Grötschel|editor3-first=László|editor3-last=Lovász|editor3-link=László Lovász|volume=I|publisher=North-Holland|year=1995}}.</ref>
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| It is one of only 13 cubic [[distance-regular graph]]s.<ref>According to the [[Foster census]].</ref>
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| The [[Graph automorphism|automorphism group]] of the Petersen graph is the [[symmetric group]] <math>S_5</math>; the action of <math>S_5</math> on the Petersen graph follows from its construction as a [[Kneser graph]]. Every [[graph homomorphism|homomorphism]] of the Petersen graph to itself that doesn't identify adjacent vertices is an automorphism. As shown in the figures, the drawings of the Petersen graph may exhibit five-way or three-way symmetry, but it is not possible to draw the Petersen graph in the plane in such a way that the drawing exhibits the full symmetry group of the graph.
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| Despite its high degree of symmetry, the Petersen graph is not a [[Cayley graph]]. It is the smallest vertex-transitive graph that is not a Cayley graph.<ref>As stated, this assumes that Cayley graphs need not be connected. Some sources require Cayley graphs to be connected, making the two-vertex [[empty graph]] the smallest vertex-transitive non-Cayley graph; under the definition given by these sources, the Petersen graph is the smallest connected vertex-transitive graph that is not Cayley.</ref>
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| == Hamiltonian paths and cycles ==
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| [[Image:Petersen2 tiny.svg|thumb|right|The Petersen graph is hypo-Hamiltonian: by deleting any vertex, such as the center vertex in the drawing, the remaining graph is Hamiltonian. This drawing with order-3 symmetry is the one given by {{harvtxt|Kempe|1886}}.]]
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| 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 graph|hypohamiltonian]], meaning that although it has no Hamiltonian cycle, deleting any vertex makes it Hamiltonian, and is the smallest hypohamiltonian graph.
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| As a finite connected vertex-transitive graph that does not have a Hamiltonian cycle, the Petersen graph is a counterexample to a variant of the [[Lovász conjecture]], but the canonical formulation of the conjecture asks for a Hamiltonian path and is verified by the Petersen graph.
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| Only five connected vertex-transitive graphs with no Hamiltonian cycles are known: the [[complete graph]] ''K''<sub>2</sub>, the Petersen graph, the [[Coxeter graph]] and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle.<ref>Royle, G. [http://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."]</ref> If ''G'' is a 2-connected, ''r''-regular graph with at most 3''r'' + 1 vertices, then ''G'' is Hamiltonian or ''G'' is the Petersen graph.<ref>{{harvtxt|Holton|Sheehan|1993}}, page 32.</ref>
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| To see that the Petersen graph has no Hamiltonian cycle ''C'', we describe the ten-vertex [[Regular graph|3-regular graphs]] that do have a Hamiltonian cycle and show that none of them is the Petersen graph, by finding a cycle in each of them that is shorter than any cycle in the Petersen graph. Any ten-vertex Hamiltonian 3-regular graph consists of a ten-vertex cycle ''C'' plus five chords. If any chord connects two vertices at distance two or three along ''C'' from each other, the graph has a 3-cycle or 4-cycle, and therefore cannot be the Petersen graph. If two chords connect opposite vertices of ''C'' to vertices at distance four along ''C'', there is again a 4-cycle. The only remaining case is a [[Möbius ladder]] formed by connecting each pair of opposite vertices by a chord, which again has a 4-cycle. Since the Petersen graph has girth five, it cannot be formed in this way and has no Hamiltonian cycle.
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| == Coloring ==
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| [[Image:PetersenBarveniHran.svg|thumb|left|A 4-coloring of the Petersen graph's edges]]
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| [[Image:Petersen graph 3-coloring.svg|thumb|right|A 3-coloring of the Petersen graph's vertices]]
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| The Petersen graph has [[chromatic number]] 3, meaning that its vertices can be [[graph coloring|colored]] with three colors — but not with two — such that no edge connects vertices of the same color.
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| The Petersen graph has [[chromatic index]] 4; coloring the edges requires four colors. A proof of this requires checking four cases to demonstrate that no 3-edge-coloring exists. As a connected bridgeless cubic graph with chromatic index four, the Petersen graph is a [[snark (graph theory)|snark]]. It is the smallest possible snark, and was the only known snark from 1898 until 1946. The [[Snark (graph theory)|snark theorem]], a result conjectured by [[W. T. Tutte]] and announced in 2001 by Robertson, Sanders, Seymour, and Thomas,<ref>{{citation|last=Pegg|first=Ed, Jr.|authorlink=Ed Pegg, Jr.|title=Book Review: The Colossal Book of Mathematics|journal=Notices of the American Mathematical Society|volume=49|issue=9|year=2002|pages=1084–1086|url=http://www.ams.org/notices/200209/rev-pegg.pdf | doi = 10.1109/TED.2002.1003756}}</ref> states that every snark has the Petersen graph as a [[Minor (graph theory)|minor]].
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| Additionally, the graph has [[fractional chromatic index]] 3, proving that the difference between the chromatic index and fractional chromatic index can be as large as 1. The long-standing Goldberg-Seymour Conjecture proposes that this is the largest gap possible.
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| The [[Thue number]] (a variant of the chromatic index) of the Petersen graph is 5.
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| == Other properties ==
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| The Petersen graph:
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| * is 3-connected and hence 3-edge-connected and bridgeless. See the [[Glossary of graph theory#Connectivity|glossary]].
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| * has independence number 4 and is 3-partite. See the [[Glossary of graph theory#Independence|glossary]].
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| * is [[cubic graph|cubic]], has [[domination number]] 3, and has a [[perfect matching]] and a [[2-factor]].
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| * has 6 distinct perfect matchings.
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| * is the smallest cubic graph of [[girth (graph theory)|girth]] 5. (It is the unique <math>(3,5)</math>-[[cage (graph theory)|cage]]. In fact, since it has only 10 vertices, it is the unique <math>(3,5)</math>-[[Moore graph]].)<ref name="hs60">{{citation
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| | author1-link = Alan Hoffman (mathematician)
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| | last1 = Hoffman
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| | first1 = Alan J.
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| | last2 = Singleton
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| | first2 = Robert R.
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| | title = Moore graphs with diameter 2 and 3
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| | journal = IBM Journal of Research and Development
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| | volume = 5
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| | issue = 4
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| | year = 1960
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| | pages = 497–504
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| | url = http://www.research.ibm.com/journal/rd/045/ibmrd0405H.pdf
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| | id = {{MathSciNet | id = 0140437}}
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| | doi=10.1147/rd.45.0497}}.</ref>
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| * has [[radius (graph theory)|radius]] 2 and [[diameter (graph theory)|diameter]] 2. It is the largest cubic graph with diameter 2.<ref>This follows from the fact that it is a Moore graph, since any Moore graph is the largest possible regular graph with its degree and diameter {{harv|Hoffman|Singleton|1960}}.</ref>
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| * has [[spectral graph theory|graph spectrum]] −2, −2, −2, −2, 1, 1, 1, 1, 1, 3.
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| * has 2000 [[Spanning tree (mathematics)|spanning tree]]s, the most of any 10-vertex cubic graph.<ref>{{harvtxt|Jakobson|Rivin|1999}}; {{harvtxt|Valdes|1991}}. The cubic graphs with 6 and 8 vertices maximizing the number of spanning trees are [[Möbius ladder]]s.</ref>
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| * has [[chromatic polynomial]] <math>t(t-1)(t-2)\left(t^7-12t^6+67t^5-230t^4+529t^3-814t^2+775t-352\right).</math><ref name="biggs">{{Citation | author=Biggs, Norman | title=Algebraic Graph Theory | edition=2nd | location=Cambridge | publisher=Cambridge University Press | year=1993 | isbn=0-521-45897-8}}</ref>
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| * has [[characteristic polynomial]] <math>(t-1)^5(t+2)^4(t-3)</math>, making it an [[integral graph]]—a graph whose [[Spectral graph theory|spectrum]] consists entirely of integers.
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| == Petersen coloring conjecture ==
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| According to DeVos, Nesetril, and Raspaud, "A ''cycle'' of a graph G is a set C <math>\subseteq</math> E(G) so that every vertex of the graph (V(G),C) has even degree. If G,H are graphs, we define a map φ: E(G) —> E(H) to be ''cycle-continuous'' if the pre-image of every cycle of H is a cycle of G. A fascinating conjecture of Jaeger asserts that every bridgeless graph has a cycle-continuous mapping to the Petersen graph. Jaeger showed that if this conjecture is true, then so are the 5-cycle-double-cover conjecture and the Berge-Fulkerson conjecture."<ref>{{citation
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| | last1 = DeVos | first1 = Matt
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| | last2 = Nešetřil | first2 = Jaroslav | author2-link = Jaroslav Nešetřil
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| | last3 = Raspaud | first3 = André
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| | contribution = On edge-maps whose inverse preserves flows or tensions
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| | doi = 10.1007/978-3-7643-7400-6_10
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| | location = Basel
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| | mr = 2279171
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| | pages = 109–138
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| | publisher = Birkhäuser
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| | series = Trends Math.
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| | title = Graph theory in Paris
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| | year = 2007}}.</ref>
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| ==Related graphs==
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| [[File:Petersen family.svg|thumb|The [[Petersen family]].]]
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| The [[generalized Petersen graph]] ''G''(''n'',''k'') is formed by connecting the vertices of a [[regular polygon|regular ''n''-gon]] to the corresponding vertices of a [[star polygon]] with [[Schläfli symbol]] {''n''/''k''}.<ref>{{harvtxt|Coxeter|1950}}; {{harvtxt|Watkins|1969}}.</ref> For instance, in this notation, the Petersen graph is ''G''(5,2): it can be formed by connecting corresponding vertices of a pentagon and five-point star, and the edges in the star connect every second vertex.
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| The generalized Petersen graphs also include the ''n''-prism ''G''(''n'',1)
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| the [[Dürer graph]] ''G''(6,2), the [[Möbius-Kantor graph]]
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| ''G''(8,3), the [[dodecahedron]] ''G''(10,2), the [[Desargues graph]] ''G''(10,3) and the [[Nauru graph]] ''G''(12,5).
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| The [[Petersen family]] consists of the seven graphs that can be formed from the Petersen graph by zero or more applications of [[Y-Δ transform|Δ-Y or Y-Δ transform]]s. The [[complete graph]] ''K''<sub>6</sub> is also in the Petersen family. These graphs form the [[forbidden minor]]s for [[linkless embedding|linklessly embeddable graphs]], graphs that can be embedded into three-dimensional space in such a way that no two cycles in the graph are [[Link (knot theory)|linked]].<ref name=Bailey1997>{{Citation|title=Surveys in Combinatorics|page=187|last1=Bailey|first1=Rosemary A.|publisher=Cambridge University Press|year=1997|isbn=978-0-521-59840-8}}</ref>
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| The [[Clebsch graph]] contains many copies of the Petersen graph as [[induced subgraph]]s: for each vertex ''v'' of the Clebsch graph, the ten non-neighbors of ''v'' induce a copy of the Petersen graph.
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| {{clear}}
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| == Notes ==
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| {{reflist|colwidth=30em}}
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| == References ==
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| {{Commons category|Petersen graph}}
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| * {{citation|first1=Geoffrey|last1=Exoo|first2=Frank|last2=Harary|authorlink2=Frank Harary|first3=Jerald|last3=Kabell|title=The crossing numbers of some generalized Petersen graphs|journal=[[Mathematica Scandinavica]]|volume=48|year=1981|pages=184–188}}.
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| *{{citation
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| | authorlink = Harold Scott MacDonald Coxeter
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| | first = H. S. M. | last = Coxeter
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| | title = Self-dual configurations and regular graphs
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| | journal = [[Bulletin of the American Mathematical Society]]
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| | volume = 56 | year = 1950 | pages = 413–455
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| | doi = 10.1090/S0002-9904-1950-09407-5
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| | issue = 5}}.
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| * {{citation|first1=D. A.|last1=Holton|first2=J.|last2=Sheehan|url=http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521435943|title=The Petersen Graph|publisher=[[Cambridge University Press]]|year=1993|isbn=0-521-43594-3|doi=10.2277/0521435943}}.
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| *{{Cite arxiv
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| | last1 = Jakobson | first1= Dmitry | last2= Rivin | first2= Igor
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| | title = On some extremal problems in graph theory
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| | year = 1999
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| | eprint= math.CO/9907050}}
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| *{{citation|first=A. B.|last=Kempe|title=A memoir on the theory of mathematical form|journal=Philosophical Transactions of the Royal Society of London|volume=177|pages=1–70|year=1886|doi=10.1098/rstl.1886.0002}}
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| * {{citation|first=László|last=Lovász|authorlink=László Lovász|title=[[Combinatorial Problems and Exercises]]|edition=2nd|publisher=North-Holland|year=1993|isbn= 0-444-81504-X}}.
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| * {{citation|first=Julius|last=Petersen|title=Sur le théorème de Tait|journal=L'Intermédiaire des Mathématiciens|volume=5|year=1898|pages=225–227}}
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| * {{cite journal
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| |first1=A. J.
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| |last1=Schwenk
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| |title=Enumeration of Hamiltonian cycles in certain generalized Petersen graphs
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| |year=1989
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| |pages=53–59
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| |journal=J. Combin. Theory B
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| |volume=47
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| |issue=1
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| |doi=10.1016/0095-8956(89)90064-6
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| }}
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| *{{cite journal
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| | last = Valdes|first= L.
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| | title = Extremal properties of spanning trees in cubic graphs
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| | journal = Congressus Numerantium
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| | year = 1991
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| | volume = 85
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| | pages = 143–160}}.
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| * {{Cite journal | first=Mark E.|last=Watkins | title=A Theorem on Tait Colorings with an Application to the Generalized Petersen Graphs | journal=[[Journal of Combinatorial Theory]] | year=1969 | volume=6 | pages=152–164 | doi=10.1016/S0021-9800(69)80116-X | issue=2}}
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| ==External links==
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| * Keller, Mitch. {{planetmath reference|id=5732|title=Kneser graphs}}
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| * {{MathWorld|urlname=PetersenGraph|title=Petersen Graph}}
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| * [http://oeis.org/search?q=Petersen+Graph&sort=&language=english&go=Search Petersen Graph] in the [[On-Line Encyclopedia of Integer Sequences]]
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| [[Category:Individual graphs]]
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| [[Category:Regular graphs]]
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