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{{Short description|Cubic graph with 10 vertices and 15 edges}}
{{Infobox graph
 | name = Petersen graph
 | image = Petersen1 tiny.svg
 | image_size = 200px
 | image_caption = The Petersen graph is most commonly drawn as a pentagon with a pentagram inside, with five spokes.
 | namesake = [[Biography:Julius Petersen|Julius Petersen]]
 | vertices = 10
 | edges = 15
 | automorphisms = 120 (S<sub>5</sub>)
 | radius = 2
 | diameter = 2
 | girth = 5
 | chromatic_number = 3
 | chromatic_index = 4
 | fractional_chromatic_index = 3
 | genus = 1
 | properties = [[Cubic graph|Cubic]]<br/>[[Strongly regular graph|Strongly regular]]<br/>[[Distance-transitive graph|Distance-transitive]]<br/>[[Snark (graph theory)|Snark]]
}}
In the [[Mathematics|mathematical]] field of [[Graph theory|graph theory]], the '''Petersen graph''' is an undirected graph with 10 [[Vertex (graph theory)|vertices]] and 15 edges. It is a small graph that serves as a useful example and [[Counterexample|counterexample]] for many problems in graph theory. The Petersen graph is named after [[Biography:Julius Petersen|Julius Petersen]], who in 1898 constructed it to be the smallest [[Bridge (graph theory)|bridgeless]] [[Cubic graph|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|author-link=Andries Brouwer}}</ref><ref>{{citation|first=Julius|last=Petersen|author-link=Julius Petersen|title=Sur le théorème de Tait|journal=[[L'Intermédiaire des Mathématiciens]]|volume=5|year=1898|pages=225&ndash;227|url=https://archive.org/details/lintermdiairede03lemogoog/page/n239/mode/1up?view=theater}}</ref>

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 |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.<ref>{{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&ndash;70|year=1886|doi=10.1098/rstl.1886.0002|s2cid=108716533 }}</ref>

[[Biography:Donald Knuth|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>

The Petersen graph also makes an appearance in [[Tropical geometry|tropical geometry]]. The cone over the Petersen graph is naturally identified with the moduli space of five-pointed rational tropical curves.

== Constructions ==
[[File:Kneser graph KG(5,2).svg|thumb|left|Petersen graph as Kneser graph <math>KG_{5,2}</math>]]
The Petersen graph is the [[Complement graph|complement]] of the [[Line graph|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|odd graph]].

Geometrically, the Petersen graph is the graph formed by the vertices and edges of the hemi-dodecahedron, that is, a [[Dodecahedron|dodecahedron]] with opposite points, lines and faces identified together.

== Embeddings ==
The Petersen graph is [[Planar graph|nonplanar]]. Any nonplanar graph has as minors either the [[Complete graph|complete graph]] <math>K_5</math>, or the [[Complete bipartite graph|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|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.

[[Image:Petersen graph, two crossings.svg|thumb|right|The Petersen graph has [[Crossing number (graph theory)|crossing number]] 2 and is [[1-planar graph|1-planar]].]]

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. Because it is nonplanar, it has at least one crossing in any drawing, and if a crossing edge is removed from any drawing it remains nonplanar and has another crossing; therefore, its crossing number is exactly 2. Each edge in this drawing is crossed at most once, so the Petersen graph is [[1-planar graph|1-planar]]. On a [[Torus|torus]] the Petersen graph can be drawn without edge crossings; it therefore has [[Genus (mathematics)|orientable genus]] 1.

[[Image:Petersen graph, unit distance.svg|thumb|right|The Petersen graph is a [[Unit distance graph|unit distance graph]]: it can be drawn in the plane with each edge having unit length.]]

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|unit distance graph]].

The simplest [[Surface (mathematics)|non-orientable surface]] on which the Petersen graph can be embedded without crossings is the [[Projective plane|projective plane]]. This is the embedding given by the hemi-dodecahedron construction of the Petersen graph (shown in the figure). The projective plane embedding can also be formed from the standard pentagonal drawing of the Petersen graph by placing a [[Cross-cap|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.

[[File:Petersen-graph.png|thumb|The Petersen graph and associated map embedded in the [[Projective plane|projective plane]]. Opposite points on the circle are identified, yielding a closed surface of non-orientable genus 1.]]

== Symmetries ==
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| 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|editor2-link=Martin Grötschel|editor3-first=László|editor3-last=Lovász|editor3-link=László Lovász| volume =I |publisher=North-Holland|year=1995| archive-url= https://web.archive.org/web/20100611212234/http://www.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps|archive-date=2010-06-11}}.</ref>
It is one of only 13 cubic [[Distance-regular graph|distance-regular graph]]s.<ref name=foster/>

The [[Graph automorphism|automorphism group]] of the Petersen graph is the [[Symmetric group|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]]. The Petersen graph is a [[Core (graph theory)|core]]: every [[Graph homomorphism|homomorphism]] of the Petersen graph to itself is an automorphism.<ref>{{citation|last=Cameron|first=Peter J.|editor1-last=Beineke|editor1-first=Lowell W.|editor2-last=Wilson|editor2-first=Robin J.|contribution=Automorphisms of graphs|doi=10.1017/CBO9780511529993|isbn=0-521-80197-4|mr=2125091|pages=135–153|publisher=Cambridge University Press, Cambridge|series=Encyclopedia of Mathematics and its Applications|title=Topics in Algebraic Graph Theory|volume=102|year=2004}}; see in particular [https://books.google.com/books?id=z2K26gZLC1MC&pg=PA153 p. 153]</ref> 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.

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.{{efn|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.}}

== Hamiltonian paths and cycles ==
[[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}}.]]

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.

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.

Only five connected vertex-transitive graphs with no Hamiltonian cycles are known: the [[Complete graph|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 name=foster>Royle, G. [http://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."] {{webarchive|url=https://web.archive.org/web/20080720005020/http://www.cs.uwa.edu.au/~gordon/remote/foster/ |date=2008-07-20 }}</ref> If ''G'' is a 2-connected, ''r''-regular graph with at most 3''r''&nbsp;+&nbsp;1 vertices, then ''G'' is Hamiltonian or ''G'' is the Petersen graph.<ref>{{citation|first1=D. A.|last1=Holton|first2=J.|last2=Sheehan|title-link= The Petersen Graph |title=The Petersen Graph|publisher={{wipe|Cambridge University Press}}|year=1993|isbn=0-521-43594-3|page=32}}</ref>

To see that the Petersen graph has no Hamiltonian cycle ''C'', consider the edges in the cut disconnecting the inner 5-cycle from the outer one. If there is a Hamiltonian cycle, an even number of these edges must be chosen. If only two of them are chosen, their end-vertices must be adjacent in the two 5-cycles, which is not possible. Hence 4 of them are chosen. Assume that the top edge of the cut is not chosen (all the other cases are the same by symmetry). Of the 5 edges in the outer cycle, the two top edges must be chosen, the two side edges must not be chosen, and hence the bottom edge must be chosen. The top two edges in the inner cycle must be chosen, but this completes a non-spanning cycle, which cannot be part of a Hamiltonian cycle. Alternatively, we can also 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.

== Coloring ==
[[Image:PetersenBarveniHran.svg|thumb|left|A 4-coloring of the Petersen graph's edges]]
[[Image:Petersen graph 3-coloring.svg|thumb|right|A 3-coloring of the Petersen graph's vertices]]

The Petersen graph has [[Chromatic number|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. It has a [[List coloring|list coloring]] with 3 colors, by Brooks' theorem for list colorings.

The Petersen graph has chromatic index 4; coloring the edges requires four colors. 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 [[Biography:W. T. Tutte|W. T. Tutte]] and announced in 2001 by Robertson, Sanders, Seymour, and Thomas,<ref>{{citation|last=Pegg|first=Ed 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=https://www.ams.org/notices/200209/rev-pegg.pdf | doi = 10.1109/TED.2002.1003756| bibcode= 2002ITED...49.1084A}}</ref> states that every snark has the Petersen graph as a minor.

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|Goldberg-Seymour Conjecture]] proposes that this is the largest gap possible.

The [[Thue number]] (a variant of the chromatic index) of the Petersen graph is 5.

The Petersen graph requires at least three colors in any (possibly improper) coloring that breaks all of its symmetries; that is, its [[Distinguishing number|distinguishing number]] is three. Except for the complete graphs, it is the only Kneser graph whose distinguishing number is not two.<ref>{{citation
 | last1 = Albertson | first1 = Michael O.
 | issue = 1
 | journal = Electronic Journal of Combinatorics
 | mr = 2285824
 | page = R20
 | title = Using determining sets to distinguish Kneser graphs
 | volume = 14
 | year = 2007| doi = 10.37236/938
 | doi-access = free
 }}.</ref>

== Other properties ==
The Petersen graph:
* is 3-connected and hence 3-edge-connected and bridgeless. See the [[Glossary of graph theory#Connectivity|glossary]].
* has independence number 4 and is 3-partite. See the [[Glossary of graph theory#Independence|glossary]].
* is [[Cubic graph|cubic]], has [[Domination number|domination number]] 3, and has a [[Perfect matching|perfect matching]] and a 2-factor.
* has 6 distinct perfect matchings.
* 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 | author1-link = Alan Hoffman (mathematician) | last1 = Hoffman | first1 = Alan J. | last2 = Singleton | first2 = Robert R.
 | title = Moore graphs with diameter 2 and 3
 | journal = IBM Journal of Research and Development
 | volume = 5
 | issue = 4
 | year = 1960
 | pages = 497–504
 | url = http://www.research.ibm.com/journal/rd/045/ibmrd0405H.pdf
 |mr=0140437
 | doi=10.1147/rd.45.0497}}.</ref>
* is the smallest cubic graph with [[Colin de Verdière graph invariant]] μ = 5.<ref>{{citation | doi=10.1090/S0002-9939-98-04244-0 | author=László Lovász, Alexander Schrijver | title=A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs | journal=[[Organization:Proceedings of the American Mathematical Society|Proceedings of the American Mathematical Society]] | volume=126 | issue=5 | year=1998 | pages=1275–1285 | s2cid=7790459 | url=https://www.ams.org/journals/proc/1998-126-05/S0002-9939-98-04244-0/S0002-9939-98-04244-0.pdf}}</ref>
* is the smallest graph of [[Cop number|cop number]] 3.<ref>{{citation
 | last1 = Baird | first1 = William
 | last2 = Beveridge | first2 = Andrew
 | last3 = Bonato | first3 = Anthony
 | last4 = Codenotti | first4 = Paolo
 | last5 = Maurer | first5 = Aaron
 | last6 = McCauley | first6 = John
 | last7 = Valeva | first7 = Silviya
 | arxiv = 1308.2841
 | issue = 1
 | journal = Contributions to Discrete Mathematics
 | mr = 3265753
 | pages = 70–84
 | title = On the minimum order of {{mvar|k}}-cop-win graphs
 | url = https://cdm.ucalgary.ca/article/view/62207
 | volume = 9
 | year = 2014
 | doi = 10.11575/cdm.v9i1.62207 | doi-access = free}}</ref>
*has [[Radius (graph theory)|radius]] 2 and [[Diameter (graph theory)|diameter]] 2. It is the largest cubic graph with diameter 2.{{efn|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.<ref name="hs60"/>}}
* has 2000 spanning trees, the most of any 10-vertex cubic graph.<ref>{{Citation
 | last1 = Jakobson | first1= Dmitry | last2= Rivin | first2= Igor
 | title = On some extremal problems in graph theory
 | year = 1999
 | arxiv= math.CO/9907050| bibcode= 1999math......7050J }}
</ref><ref>{{citation| last = Valdes|first= L.| title = Extremal properties of spanning trees in cubic graphs
 | journal = Congressus Numerantium
 | year = 1991
 | volume = 85
 | pages = 143–160}}.
</ref>{{efn|The cubic graphs with 6 and 8 vertices maximizing the number of spanning trees are [[Möbius ladder]]s.}}
* has [[Chromatic polynomial|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>
* has [[Characteristic polynomial|characteristic polynomial]] <math>(t-1)^5(t+2)^4(t-3)</math>, making it an [[Integral graph|integral graph]]—a graph whose [[Spectral graph theory|spectrum]] consists entirely of integers.

== Petersen coloring conjecture ==
An ''Eulerian subgraph'' of a graph <math>G</math> is a subgraph consisting of a subset of the edges of <math>G</math>, touching every vertex of <math>G</math> an even number of times. These subgraphs are the elements of the [[Cycle space|cycle space]] of <math>G</math> and are sometimes called cycles. If <math>G</math> and <math>H</math> are any two graphs, a function from the edges of <math>G</math> to the edges of <math>H</math> is defined to be ''cycle-continuous'' if the pre-image of every cycle of <math>H</math> is a cycle of <math>G</math>. A conjecture of Jaeger asserts that every bridgeless graph has a cycle-continuous mapping to the Petersen graph. Jaeger showed this conjecture implies the 5-[[Cycle double cover|cycle-double-cover]] conjecture and the Berge-Fulkerson conjecture."<ref>{{citation| last1 = DeVos | first1 = Matt
 | last3 = Raspaud | first3 = André
 | contribution = On edge-maps whose inverse preserves flows or tensions
 | doi = 10.1007/978-3-7643-7400-6_10
 | location = Basel
 | mr = 2279171
 | pages = 109–138
 | publisher = Birkhäuser
 | series = Trends Math.
 | title = Graph theory in Paris
 | year = 2007}}.</ref>

==Related graphs==
[[File:Petersen family.svg|thumb|The [[Petersen family|Petersen family]].]]
The [[Generalized Petersen graph|generalized Petersen graph]] <math>G(n,k)</math> is formed by connecting the vertices of a [[Regular polygon|regular ''n''-gon]] to the corresponding vertices of a [[Star polygon|star polygon]] with [[Schläfli symbol]] {''n''/''k''}.<ref>{{citation | author-link = Harold Scott MacDonald Coxeter
 | first = H. S. M. | last = Coxeter
 | title = Self-dual configurations and regular graphs
 | journal = [[Organization:Bulletin of the American Mathematical Society|Bulletin of the American Mathematical Society]]
 | volume = 56 | year = 1950 | pages = 413–455
 | doi = 10.1090/S0002-9904-1950-09407-5
 | issue = 5| doi-access = free
 }}.
</ref><ref>{{Citation | 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&ndash;164 | doi=10.1016/S0021-9800(69)80116-X | issue=2| doi-access=free }}</ref> For instance, in this notation, the Petersen graph is <math>G(5,2)</math>: 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. The generalized Petersen graphs also include the ''n''-prism <math>G(n,1)</math> the [[Dürer graph]] <math>G(6,2)</math>, the Möbius-Kantor graph <math>G(8,3)</math>, the [[Dodecahedron|dodecahedron]] <math>G(10,2)</math>, the [[Desargues graph]] <math>G(10,3)</math> and the [[Nauru graph]] <math>G(12,5)</math>.

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|complete graph]] ''K''<sub>6</sub> is also in the Petersen family. These graphs form the forbidden minors 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>

The [[Clebsch graph]] contains many copies of the Petersen graph as [[Induced subgraph|induced subgraph]]s: for each vertex ''v'' of the Clebsch graph, the ten non-neighbors of ''v'' induce a copy of the Petersen graph.

{{clear}}

==Notes==
{{notelist}}

==References==
{{reflist}}

==Further reading==
* {{citation|first1=Geoffrey|last1=Exoo|first2=Frank|last2=Harary|first3=Jerald|last3=Kabell|title=The crossing numbers of some generalized Petersen graphs|journal=Mathematica Scandinavica|volume=48|year=1981|pages=184&ndash;188|doi=10.7146/math.scand.a-11910|doi-access=free}}.
* {{citation|first=László|last=Lovász|title=Combinatorial Problems and Exercises|edition=2nd|publisher=North-Holland|year=1993|isbn= 0-444-81504-X}}.
* {{citation
|first1=A. J.
|last1=Schwenk
|title=Enumeration of Hamiltonian cycles in certain generalized Petersen graphs
|year=1989
|pages=53&ndash;59
|journal=Journal of Combinatorial Theory|series=Series B
|volume=47
|issue=1
|doi=10.1016/0095-8956(89)90064-6
|doi-access=free
}}

==External links==
* {{MathWorld|urlname=PetersenGraph|title=Petersen Graph|mode=cs2}}
* [http://oeis.org/search?q=Petersen+Graph&sort=&language=english&go=Search Petersen Graph] in the [[On-Line Encyclopedia of Integer Sequences]]

[[Category:Individual graphs]]
[[Category:Regular graphs]]
[[Category:Strongly regular graphs]]

{{Sourceattribution|Petersen graph}}