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| In [[combinatorics|combinatorial]] [[mathematics]], the '''Albertson conjecture''' is an unproven relationship between the [[Crossing number (graph theory)|crossing number]] and the [[chromatic number]] of a [[graph (mathematics)|graph]]. It is named after Michael O. Albertson, a professor at [[Smith College]], who stated it as a conjecture in 2007;<ref>According to {{harvtxt|Albertson|Cranston|Fox|2009}}, the conjecture was made by Albertson at a special session of the [[American Mathematical Society]] in Chicago, held in October 2007.</ref> it is one of his many conjectures in [[graph coloring]] theory.<ref>{{citation|title=In memory of Michael O. Albertson, 1946–2009: a collection of his outstanding conjectures and questions in graph theory|first=Joan P.|last=Hutchinson|authorlink=Joan Hutchinson|date=June 19, 2009|url=http://orion.math.iastate.edu/rymartin/dm-net/MOASIAM.pdf|publisher=SIAM Activity group on Discrete Mathematics}}.</ref> The conjecture states that, among all graphs requiring ''n'' colors, the [[complete graph]] ''K''<sub>''n''</sub> is the one with the smallest crossing number.
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| Equivalently, if a graph can be drawn with fewer crossings than ''K''<sub>''n''</sub>, then, according to the conjecture, it may be colored with fewer than ''n'' colors.
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| ==A conjectured formula for the minimum crossing number==
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| It is straightforward to show that graphs with bounded crossing number have bounded chromatic number: one may assign distinct colors to the endpoints of all crossing edges and then 4-color the remaining [[planar graph]]. Albertson's conjecture replaces this qualitative relationship between crossing number and coloring by a more precise quantitative relationship. Specifically,
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| a different conjecture of {{harvs|first=Richard K.|last=Guy|authorlink=Richard K. Guy|year=1972|txt}} states that the crossing number of the complete graph ''K''<sub>''n''</sub> is
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| :<math>\textrm{cr}(K_n)=\frac14\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor.</math>
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| It is known how to draw complete graphs with this many crossings, by placing the vertices in two concentric circles; what is unknown is whether there exists a better drawing with fewer crossings. Therefore, a strengthened formulation of the Albertson conjecture is that every ''n''-chromatic graph has crossing number at least as large as the right hand side of this formula.<ref name="acf09">{{harvtxt|Albertson|Cranston|Fox|2009}}.</ref> This strengthened conjecture would be true if and only if both Guy's conjecture and the Albertson conjecture are true.
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| ==Asymptotic bounds==
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| A weaker form of the conjecture, proven by M. Schaefer,<ref name="acf09"/> states that every graph with chromatic number ''n'' has crossing number Ω(''n''<sup>4</sup>), or equivalently that every graph with crossing number ''k'' has chromatic number ''O''(''k''<sup>1/4</sup>). {{harvtxt|Albertson|Cranston|Fox|2009}} published a simple proof of these bounds, by combining the fact that every ''n''-chromatic graph has minimum degree at least ''n'' (else it would have a [[greedy coloring]] with fewer colors) together with the [[Crossing number (graph theory)|crossing number inequality]] according to which every graph ''G'' = (''V'',''E'') with |''E''|/|''V''| ≥ 4 has crossing number Ω(|''E''|<sup>3</sup>/|''V''|<sup>2</sup>). Using the same reasoning, they show that a counterexample to Albertson's conjecture for the chromatic number ''n'' (if it exists) must have fewer than 4''n'' vertices.
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| ==Special cases==
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| The Albertson conjecture is [[vacuous truth|vacuously true]] for ''n'' ≤ 4: ''K''<sub>4</sub> has crossing number zero, and all graphs have crossing number greater than or equal to zero. The case ''n'' = 5 of Albertson's conjecture is equivalent to the [[four color theorem]], that any planar graph can be colored with four or fewer colors, for the only graphs requiring fewer crossings than the one crossing of ''K''<sub>5</sub> are the planar graphs, and the conjecture implies that these should all be at most 4-chromatic. Through the efforts of several groups of authors the conjecture is now known to hold for all ''n'' ≤ 16.<ref>{{harvtxt|Oporowski|Zhao|2009}}; {{harvtxt|Albertson|Cranston|Fox|2009}}; {{harvtxt|Barát|Tóth|2010}}.</ref>
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| ==Related conjectures==
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| There is also a connection to the [[Hadwiger conjecture (graph theory)|Hadwiger conjecture]], an important open problem in combinatorics concerning the relationship between chromatic number and the existence of large [[clique (graph theory)|cliques]] as [[graph minor|minors]] in a graph.<ref>{{harvtxt|Barát|Tóth|2009}}.</ref> A variant of the Hadwiger conjecture, stated by [[György Hajós]], is that every ''n''-chromatic graph contains a [[Homeomorphism (graph theory)|subdivision]] of ''K''<sub>''n''</sub>; if this were true, the Albertson conjecture would follow, because the crossing number of the whole graph is at least as large as the crossing number of any of its subdivisions. However, counterexamples to the Hajós conjecture are now known,<ref>{{harvtxt|Catlin|1979}}; {{harvtxt|Erdős|Fajtlowicz|1981}}.</ref> so this connection does not provide an avenue for proof of the Albertson conjecture.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|first1=Michael O.|last1=Albertson|first2=Daniel W.|last2=Cranston|first3=Jacob|last3=Fox|title=Colorings, crossings, and cliques|journal=Electronic Journal of Combinatorics|volume=16|year=2009|pages=R45|url=http://www.combinatorics.org/Volume_16/PDF/v16i1r45.pdf|arxiv=1006.3783}}.
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| *{{citation|first1=János|last1=Barát|first2=Géza|last2=Tóth|year=2010|title=Towards the Albertson Conjecture|arxiv=0909.0413|journal=Electronic Journal of Combinatorics|volume=17|issue=1|page=R73|url=http://www.combinatorics.org/Volume_17/Abstracts/v17i1r73.html}}.
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| *{{citation | last = Catlin | first = P. A. | title = Hajós's graph-colouring conjecture: variations and counterexamples | journal = Journal of Combinatorial Theory, Series B | volume = 26 | year = 1979 | issue = 2 | pages = 268–274 | doi = 10.1016/0095-8956(79)90062-5}}.
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| *{{citation | last1 = Erdős | first1 = Paul | authorlink1 = Paul Erdős | last2 = Fajtlowicz | first2 = Siemion | author2-link=Siemion Fajtlowicz|title = On the conjecture of Hajós | journal = [[Combinatorica]] | volume = 1 | issue = 2 | year = 1981 | pages = 141–143 | doi = 10.1007/BF02579269}}.
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| *{{citation|first=Richard K.|last=Guy|authorlink=Richard K. Guy|year=1972|contribution=Crossing numbers of graphs|title=Graph Theory and Applications: Proceedings of the Conference at Western Michigan University, Kalamazoo, Mich., May 10–13, 1972|editor1-first=Y.|editor1-last=Alavi|editor1-link=Yousef Alavi|editor2-first=D. R.|editor2-last=Lick|editor3-first=A. T.|editor3-last=White|location=New York|publisher=Springer-Verlag|pages=111–124}}. As cited by {{harvtxt|Albertson|Cranston|Fox|2009}}.
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| *{{citation|first1=B.|last1=Oporowski|first2=D.|last2=Zhao|title=Coloring graphs with crossings|arxiv=math/0501427|year=2009|journal=Discrete Mathematics|volume=309|issue=9|pages=2948–2951|doi=10.1016/j.disc.2008.07.040}}.
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| [[Category:Topological graph theory]]
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| [[Category:Graph coloring]]
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| [[Category:Conjectures]]
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