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| {{distinguish2|[[Graph of a function|graphs]] of [[cubic function]]s}}
| | Very first scratch . know exactly what a new video game world could have. There are horrors with bad in every place and cranny. This amazing post includes advice related to optimizing your gaming time with tricks and hints you might not be a little more aware of. Embark on reading for more important info.<br><br>If you have got to reload one specific arms when playing deviate of clans hack that has shooting entailed, always get cover first. This process is common for guitar players to be gunned back down while a reload might be happening, and you wrist watch helplessly. Do Not even let it happen you! Find somewhere that will conceal before you get started to reload.<br><br>Nevertheless, if you want avoid at the top of the competitors, there are a few simple points you truly keep in mind. Realize your foe, understand game and the win will be yours. It is possible in order to consider the aid of clash of clans hack tools and supplementary rights if you just like your course. Terribly for your convenience, [http://Beneath.com/ beneath] are the general details in this sport that you will need remember of. Go through all of them carefully!<br><br>In the event the system that your a person is enjoying on may want to connect with the Net, be sure that you'll fix the settings for your loved ones before he performs by it. You're going to be able to safeguard your kid brought on by vulnerability to unsavory written content utilizing these filter ways. There are also options to establish the amount of discussion they can participate with other individuals when online.<br><br>Coursesmart not only provides overall tools, there is usually [http://circuspartypanama.com Clash of Clans hack no survey] by anyone. Strict anti ban system take users to utilize this program and play without much hindrance. If players are interested in obtaining the program, they are obviously required to visit amazing site and obtain the most important hack tool trainer at this instant. The name of the online site is Amazing Cheats. A number of web stores have different types from software by which many can get past a difficult situation stages in the task.<br><br>Rare metal and Elixir would work as the main sources available about Clash of Clans. Each of these regarding are necessary and can be gathered by a volume of ways. Frontrunners of the people can use structures, recover the cash some other tribes actually clash of [http://www.squidoo.com/search/results?q=clans+crack clans crack] tools for acquiring them both.<br><br>There is an helpful component of this particular diversion as fantastic. When one particular player has modified, the Battle of Clan Castle damages in his or lady's village, he or she could successfully start or subscribe to for each faction using diverse gamers exactly where they can take a look at with every other current troops to just each other well these troops could get in touch either offensively or protectively. The Clash from Clans cheat for free additionally holds the greatest district centered globally converse so gamers could laps making use of many types of players for social courting and as faction joining.This recreation is a have to to play on your android application specially if you typically employing my clash relating to clans android hack resource. |
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| [[Image:Petersen1 tiny.svg|thumb|Right|The [[Petersen graph]] is a Cubic graph.]]
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| [[Image:Biclique K 3 3.svg|thumb|180px|Right|The [[complete bipartite graph]] <math>K_{3,3}</math> is an example of a bicubic graph]]
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| In the [[mathematics|mathematical]] field of [[graph theory]], a '''cubic graph''' is a [[graph (mathematics)|graph]] in which all [[vertex (graph theory)|vertices]] have [[degree (graph theory)|degree]] three. In other words a cubic graph is a 3-[[regular graph]]. Cubic graphs are also called '''trivalent graphs'''.
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| A '''bicubic graph''' is a cubic [[bipartite graph]].
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| ==Symmetry==
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| In 1932, [[R. M. Foster|Ronald M. Foster]] began collecting examples of cubic [[symmetric graph]]s, forming the start of the [[Foster census]].<ref name="Ref_Foster">{{Citation|first1=R. M.|last1=Foster|title=Geometrical Circuits of Electrical Networks|journal=[[Transactions of the American Institute of Electrical Engineers]]|volume=51|pages=309–317|year=1932|doi=10.1109/T-AIEE.1932.5056068|issue=2}}.</ref> Many well-known individual graphs are cubic and symmetric, including the [[Water, gas, and electricity|utility graph]], the [[Petersen graph]], the [[Heawood graph]], the [[Möbius–Kantor graph]], the [[Pappus graph]], the [[Desargues graph]], the [[Nauru graph]], the [[Coxeter graph]], the [[Tutte–Coxeter graph]], the [[Dyck graph]], the [[Foster graph]] and the [[Biggs-Smith graph]]. [[W. T. Tutte]] classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with each possible value of ''s'' from 1 to 5.<ref>{{Citation
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| | doi = 10.4153/CJM-1959-057-2
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| | last = Tutte | first = W. T.
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| | journal = Canad. J. Math
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| | pages = 621–624
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| | title = On the symmetry of cubic graphs
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| | url = http://cms.math.ca/cjm/v11/p621
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| | volume = 11
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| | year = 1959}}.</ref>
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| [[Semi-symmetric graph|Semi-symmetric]] cubic graphs include the [[Gray graph]] (the smallest semi-symmetric cubic graph), the [[Ljubljana graph]], and the [[Tutte 12-cage]].
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| The [[Frucht graph]] is one of the two smallest cubic graphs without any symmetries: it possesses only a single [[graph automorphism]], the identity automorphism.
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| ==Coloring and independent sets==
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| According to [[Brooks' theorem]] every cubic graph other than the [[complete graph]] ''K''<sub>4</sub> can be [[graph coloring|colored]] with at most three colors. Therefore, every cubic graph other than ''K''<sub>4</sub> has an [[independent set (graph theory)|independent set]] of at least ''n''/3 vertices, where ''n'' is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices.
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| According to [[Vizing's theorem]] every cubic graph needs either three or four colors for an edge coloring. A 3-edge-coloring is known as a Tait coloring, and forms a partition of the edges of the graph into three [[perfect matching]]s. By [[König's theorem (graph theory)|König's line coloring theorem]] every bicubic graph has a Tait coloring.
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| The bridgeless cubic graphs that do not have a Tait coloring are known as [[Snark (graph theory)|snarks]]. They include the [[Petersen graph]], [[Tietze's graph]], the [[Blanuša snarks]], the [[flower snark]], the [[double-star snark]], the [[Szekeres snark]] and the [[Watkins snark]]. There is an infinite number of distinct snarks.<ref>{{citation
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| | doi = 10.2307/2319844
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| | last = Isaacs | first = R.
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| | issue = 3
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| | journal = [[American Mathematical Monthly]]
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| | pages = 221–239
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| | title = Infinite families of nontrivial trivalent graphs which are not Tait colorable
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| | jstor = 2319844
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| | volume = 82
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| | year = 1975}}.</ref>
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| ==Topology and geometry==
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| Cubic graphs arise naturally in [[topology]] in several ways. For example, if one considers a [[Graph (mathematics)|graph]] to be a 1-dimensional [[CW complex]], cubic graphs are ''generic'' in that most 1-cell attaching maps are disjoint from the 0-skeleton of the graph. Cubic graphs are also formed as the graphs of [[polyhedron|simple polyhedra]] in three dimensions, polyhedra such as the [[regular dodecahedron]] with the property that three faces meet at every vertex.
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| An arbitrary [[graph embedding]] on a two-dimensional surface may be represented as a cubic graph structure known as a [[graph-encoded map]]. In this structure, each vertex of a cubic graph represents a [[Flag (geometry)|flag]] of the embedding, a mutually incident triple of a vertex, edge, and face of the surface. The three neighbors of each flag are the three flags that may be obtained from it by changing one of the members of this mutually incident triple and leaving the other two members unchanged.<ref>{{citation
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| | last1 = Bonnington | first1 = C. Paul
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| | last2 = Little | first2 = Charles H. C.
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| | publisher = Springer-Verlag
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| | title = The Foundations of Topological Graph Theory
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| | year = 1995}}.</ref>
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| ==Hamiltonicity==
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| There has been much research on [[Hamiltonian cycle|Hamiltonicity]] of cubic graphs. In 1880, [[Peter Guthrie Tait|P.G. Tait]] conjectured that every cubic [[polyhedral graph]] has a [[Hamiltonian circuit]]. [[William Thomas Tutte]] provided a counter-example to [[Tait's conjecture]], the 46-vertex [[Tutte graph]], in 1946. In 1971, Tutte conjectured that all bicubic graphs are Hamiltonian. However, Joseph Horton provided a counterexample on 96 vertices, the [[Horton graph]].<ref name="Ref_a">Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 240, 1976.</ref> Later, Mark Ellingham constructed two more counterexamples : the [[Ellingham-Horton graph]]s.<ref name="Ref_b">Ellingham, M. N. "Non-Hamiltonian 3-Connected Cubic Partite Graphs."Research Report No. 28, Dept. of Math., Univ. Melbourne, Melbourne, 1981.</ref><ref name="Ref_c">{{Citation|last1=Ellingham|first1=M. N.|last2=Horton|first2=J. D.|title= Non-Hamiltonian 3-connected cubic bipartite graphs|journal=[[Journal of Combinatorial Theory]]|series=Series B| volume=34| pages=350–353| year=1983| doi=10.1016/0095-8956(83)90046-1|issue=3}}.</ref> [[Barnette's conjecture]], a still-open combination of Tait's and Tutte's conjecture, states that every bicubic polyhedral graph is Hamiltonian. When a cubic graph is Hamiltonian, [[LCF notation]] allows it to be represented concisely.
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| If a cubic graph is chosen [[random graph|uniformly at random]] among all ''n''-vertex cubic graphs, then it is very likely to be Hamiltonian: the proportion of the ''n''-vertex cubic graphs that are Hamiltonian tends to one in the limit as ''n'' goes to infinity.<ref>{{citation
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| | last1 = Robinson | first1 = R.W.
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| | last2 = Wormald | first2 = N.C.
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| | doi = 10.1002/rsa.3240050209
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| | issue = 2
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| | journal = Random Structures and Algorithms
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| | pages = 363–374
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| | title = Almost all regular graphs are Hamiltonian
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| | volume = 5
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| | year = 1994}}.</ref>
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| [[David Eppstein]] conjectured that every ''n''-vertex cubic graph has at most 2<sup>''n''/3</sup> (approximately 1.260<sup>''n''</sup>) distinct Hamiltonian cycles, and provided examples of cubic graphs with that many cycles.<ref>{{citation
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| | last = Eppstein | first = David | authorlink = David Eppstein
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| | arxiv = cs.DS/0302030 | issue = 1
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| | journal = [[Journal of Graph Algorithms and Applications]]
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| | pages = 61–81
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| | title = The traveling salesman problem for cubic graphs
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| | url = http://jgaa.info/accepted/2007/Eppstein2007.11.1.pdf
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| | volume = 11
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| | year = 2007}}.</ref> The best upper bound that has been proven on the number of distinct Hamiltonian cycles is 1.276<sup>''n''</sup>.<ref>{{citation
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| | last = Gebauer | first = H.
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| | contribution = On the number of Hamilton cycles in bounded degree graphs
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| | title = Proc. 4th Workshop on Analytic Algorithmics and Combinatorics (ANALCO '08)
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| | url = http://zeno.siam.org/proceedings/analco/2008/anl08_023gebauerh.pdf
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| | year = 2008}}.</ref>
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| ==Other properties==
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| The [[pathwidth]] of any ''n''-vertex cubic graph is at most ''n''/6. However, the best known lower bound on the pathwidth of cubic graphs is smaller, 0.082''n''.<ref name="fh06">{{citation
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| | last1 = Fomin | first1 = Fedor V.
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| | last2 = Høie | first2 = Kjartan
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| | doi = 10.1016/j.ipl.2005.10.012
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| | issue = 5
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| | journal = [[Information Processing Letters]]
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| | pages = 191–196
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| | title = Pathwidth of cubic graphs and exact algorithms
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| | volume = 97
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| | year = 2006}}.</ref>
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| It follows from the [[handshaking lemma]], proven by [[Leonhard Euler]] in 1736 as part of the first paper on graph theory, that every cubic graph has an even number of vertices.
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| [[Petersen's theorem]] states that every cubic [[bridge (graph theory)|bridgeless]] graph has a [[perfect matching]].<ref name="Pet1891">{{Citation
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| | last1 = Petersen | first1 = Julius Peter Christian
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| | issue = 15
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| | journal = [[Acta Mathematica]]
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| | pages = 193–220
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| | title = Die Theorie der regulären Graphs (The theory of regular graphs)
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| | doi=10.1007/BF02392606
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| | year = 1891
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| | volume = 15}}.</ref>
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| [[László Lovász|Lovász]] and [[Michael D. Plummer|Plummer]] conjectured that every cubic bridgeless graph has an exponential number of perfect matchings. The conjecture was recently proved, showing that every cubic bridgeless graph with ''n'' vertices has at least 2<sup>n/3656</sup> perfect matchings.<ref name="EKKKN11">{{citation
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| | last1 = Esperet | first1 = Louis
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| | last2 = Kardoš | first2 = František
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| | last3 = King | first3 = Andrew D.
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| | last4 = Král | first4 = Daniel
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| | last5 = Norine | first5 = Serguei
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| | doi = 10.1016/j.aim.2011.03.015
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| | issue = 4
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| | journal = [[Advances in Mathematics]]
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| | pages = 1646–1664
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| | title = Exponentially many perfect matchings in cubic graphs
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| | year = 2011
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| | volume = 227}}.</ref>
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| ==Algorithms and complexity==
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| Several researchers have studied the complexity of [[exponential time]] algorithms restricted to cubic graphs. For instance, by applying [[dynamic programming]] to a [[path decomposition]] of the graph, Fomin and Høie showed how to find their [[maximum independent set]]s in time O(2<sup>''n''/6 + o(n)</sup>).<ref name="fh06"/> The [[travelling salesman problem]] can be solved in cubic graphs in time O(1.251<sup>''n''</sup>).<ref name="Iwama2007">{{citation|first1=Kazuo|last1=Iwama|first2=Takuya|last2=Nakashima|series=Lecture Notes in Computer Science|publisher=Springer-Verlag|title=Computing and Combinatorics|contribution=An Improved Exact Algorithm for Cubic Graph TSP|year=2007|doi=10.1007/978-3-540-73545-8_13|volume=4598|pages=108–117|isbn=978-3-540-73544-1}}.</ref>
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| Several important graph optimization problems are [[APX|APX hard]], meaning that, although they have [[approximation algorithm]]s whose [[approximation ratio]] is bounded by a constant, they do not have [[polynomial time approximation scheme]]s whose approximation ratio tends to 1 unless [[P vs NP problem|P=NP]]. These include the problems of finding a minimum [[vertex cover]], [[maximum independent set]], minimum [[dominating set]], and [[maximum cut]].<ref>{{citation
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| | last1 = Alimonti | first1 = Paola
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| | last2 = Kann | first2 = Viggo
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| | doi = 10.1016/S0304-3975(98)00158-3
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| | issue = 1–2
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| | journal = [[Theoretical Computer Science (journal)|Theoretical Computer Science]]
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| | pages = 123–134
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| | title = Some APX-completeness results for cubic graphs
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| | volume = 237
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| | year = 2000}}.</ref>
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| The [[Crossing number (graph theory)|crossing number]] (the minimum number of edges which cross in any [[graph drawing]]) of a cubic graph is also [[NP-hard]] for cubic graphs but may be approximated.<ref name="Hlinny2006">{{citation|first=Petr|last=Hliněný|title=Crossing number is hard for cubic graphs|journal=[[Journal of Combinatorial Theory]]|series=Series B|volume=96|issue=4|pages=455–471|year=2006|doi=10.1016/j.jctb.2005.09.009}}.</ref>
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| The [[Travelling Salesman Problem]] on cubic graphs has been proven to be [[NP-hard]] to approximate to within any factor less than 1153/1152.<ref>{{citation
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| | first1 = Marek | last1 = Karpinski
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| | first2 = Richard | last2 = Schmied
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| | arxiv = 1304.6800
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| | title = Approximation Hardness of Graphic TSP on Cubic Graphs
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| | year = 2013}}.</ref>
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| ==See also==
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| * [[Table of simple cubic graphs]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| *{{ cite web|url=http://mapleta.maths.uwa.edu.au/~gordon/remote/foster/|first1=Gordon|last1=Royle|title=Cubic symmetric graphs (The Foster Census)
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| }}
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| *{{mathworld|urlname=BicubicGraph|title=Bicubic Graph}}
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| *{{mathworld|urlname=CubicGraph|title=Cubic Graph}}
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| [[Category:Graph families]]
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| [[Category:Regular graphs]]
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