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| David Sumner (a [[graph theory|graph theorist]] at the [[University of South Carolina]]) [[conjecture]]d in 1971 that [[tournament (graph theory)|tournaments]] are [[universal graph]]s for [[polytree]]s. More precisely, '''Sumner's conjecture''' (also called '''Sumner's universal tournament conjecture''') states that every [[Orientation (graph theory)|orientation]] of every <math>n</math>-vertex [[Tree (graph theory)|tree]] is a [[subgraph]] of every <math>(2n-2)</math>-vertex tournament.<ref>{{harvtxt|Kühn|Mycroft|Osthus|2011a}}. However the earliest published citations given by Kühn et al. are to {{harvtxt|Reid|Wormald|1983}} and {{harvtxt|Wormald|1983}}. {{harvtxt|Wormald|1983}} cites the conjecture as an undated private communication by Sumner.</ref> The conjecture remains unproven; {{harvtxt|Kühn|Mycroft|Osthus|2011a}} call it "one of the most well-known problems on tournaments."
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| ==Examples==
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| Let polytree <math>P</math> be a [[Star (graph theory)|star]] <math>K_{1,n-1}</math>, in which all edges are oriented outward from the central vertex to the leaves. Then, <math>P</math> cannot be embedded in the tournament formed from the vertices of a regular <math>2n-3</math>-gon by directing every edge clockwise around the polygon. For, in this tournament, every vertex has indegree and outdegree equal to <math>n-2</math>, while the central vertex in <math>P</math> has larger outdegree <math>n-1</math>.<ref>This example is from {{harvtxt|Kühn|Mycroft|Osthus|2011a}}.</ref> Thus, if true, Sumner's conjecture would give the best possible size of a universal graph for polytrees.
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| However, in every tournament of <math>2n-2</math> vertices, the average outdegree is <math>n-\frac{3}{2}</math>, and the maximum outdegree is an integer greater than or equal to the average. Therefore, there exists a vertex of outdegree <math>\left\lceil n-\frac{3}{2}\right\rceil=n-1</math>, which can be used as the central vertex for a copy of <math>P</math>.
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| ==Partial results==
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| The following partial results on the conjecture are known. | |
| *It is true for all sufficiently large values of <math>n</math>.<ref>{{harvtxt|Kühn|Mycroft|Osthus|2011b}}.</ref>
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| *There is a function <math>f(n)</math> with asymptotic growth rate <math>f(n)=2n+o(n)</math> with the property that every <math>n</math>-vertex polytree can be embedded as a subgraph of every <math>f(n)</math>-vertex tournament. Additionally and more explicitly, <math>f(n)\le 3n-3</math>.<ref>{{harvtxt|Kühn|Mycroft|Osthus|2011a}} and {{harvtxt|El Sahili|2004}}. For earlier weaker bounds on <math>f(n)</math>, see {{harvtxt|Chung|1981}}, {{harvtxt|Wormald|1983}}, {{harvtxt|Häggkvist|Thomason|1991}}, {{harvtxt|Havet|Thomassé|2000b}}, and {{harvtxt|Havet|2002}}.</ref>
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| *There is a function <math>g(k)</math> such that tournaments on <math>n+g(k)</math> vertices are universal for polytrees with <math>k</math> leaves.<ref>{{harvtxt|Häggkvist|Thomason|1991}}; {{harvtxt|Havet|Thomassé|2000a}}; {{harvtxt|Havet|2002}}.</ref>
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| *There is a function <math>h(n,\Delta)</math> such that every <math>n</math>-vertex polytree with maximum degree at most <math>\Delta</math> forms a subgraph of every tournament with <math>h(n,\Delta)</math> vertices. When <math>\Delta</math> is a fixed constant, the asymptotic growth rate of <math>h(n,\Delta)</math> is <math>n+o(n)</math>.<ref>{{harvtxt|Kühn|Mycroft|Osthus|2011a}}.</ref>
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| *Every "near-regular" tournament on <math>2n-2</math> vertices contains every <math>n</math>-vertex polytree.<ref name="rw83">{{harvtxt|Reid|Wormald|1983}}.</ref>
| |
| *Every orientation of an <math>n</math>-vertex [[caterpillar tree]] with [[diameter (graph theory)|diameter]] at most four can be embedded as a subgraph of every <math>(2n-2)</math>-vertex tournament.<ref name="rw83"/>
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| *Every <math>(2n-2)</math>-vertex tournament contains as a subgraph every <math>n</math>-vertex [[rooted tree]].<ref>{{harvtxt|Havet|Thomassé|2000b}}.</ref>
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| ==Related conjectures==
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| {{harvtxt|Rosenfeld|1972}} conjectured that every orientation of an <math>n</math>-vertex [[path graph]] (with <math>n\ge 8</math>) can be embedded as a subgraph into every <math>n</math>-vertex tournament.<ref name="rw83"/> After partial results by {{harvtxt|Thomason|1986}} this was proven by {{harvtxt|Havet|Thomassé|2000a}}.
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| Havet and Thomassé<ref>In {{harvtxt|Havet|2002}}, but jointly credited to Thomassé in that paper.</ref> in turn conjectured a strengthening of Sumner's conjecture, that every tournament on <math>n+k-1</math> vertices contains as a subgraph every polytree with at most <math>k</math> leaves.
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| {{harvtxt|Burr|1980}} conjectured that, whenever a graph <math>G</math> requires <math>2n-2</math> or more colors in a [[graph coloring|coloring]] of <math>G</math>, then every orientation of <math>G</math> contains every orientation of an <math>n</math>-vertex tree. Because complete graphs require a different color for each vertex, Sumner's conjecture would follow immediately from Burr's conjecture.<ref>This is a corrected version of Burr's conjecture from {{harvtxt|Wormald|1983}}.</ref> As Burr showed, orientations of graphs whose chromatic number grows quadratically as a function of <math>n</math> are universal for polytrees.
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| ==Notes==
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| {{reflist|colwidth=30em}}
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| ==References==
| |
| *{{citation
| |
| | last = Burr | first = Stefan A. | author-link = Stefan Burr
| |
| | contribution = Subtrees of directed graphs and hypergraphs
| |
| | series = Congressus Numerantium
| |
| | mr = 608430
| |
| | pages = 227–239
| |
| | title = Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1980), Vol. I
| |
| | volume = 28
| |
| | year = 1980}}.
| |
| *{{citation
| |
| | last = Chung | first = F.R.K. | author-link = Fan Chung
| |
| | publisher = [[Bell Laboratories]]
| |
| | series = Internal Memorandum
| |
| | title = A note on subtrees in tournaments
| |
| | year = 1981}}. As cited by {{harvtxt|Wormald|1983}}.
| |
| *{{citation
| |
| | last = El Sahili | first = A.
| |
| | doi = 10.1016/j.jctb.2004.04.002
| |
| | issue = 1
| |
| | journal = [[Journal of Combinatorial Theory]] | series = Series B
| |
| | mr = 2078502
| |
| | pages = 183–187
| |
| | title = Trees in tournaments
| |
| | volume = 92
| |
| | year = 2004}}.
| |
| *{{citation
| |
| | last1 = Häggkvist | first1 = Roland
| |
| | last2 = Thomason | first2 = Andrew
| |
| | doi = 10.1007/BF01206356
| |
| | issue = 2
| |
| | journal = [[Combinatorica]]
| |
| | mr = 1136161
| |
| | pages = 123–130
| |
| | title = Trees in tournaments
| |
| | volume = 11
| |
| | year = 1991}}.
| |
| *{{citation
| |
| | last = Havet | first = Frédéric
| |
| | doi = 10.1016/S0012-365X(00)00463-5
| |
| | issue = 1-3
| |
| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| |
| | mr = 1874730
| |
| | pages = 121–134
| |
| | title = Trees in tournaments
| |
| | volume = 243
| |
| | year = 2002}}.
| |
| *{{citation | |
| | last1 = Havet | first1 = Frédéric
| |
| | last2 = Thomassé | first2 = Stéphan
| |
| | doi = 10.1006/jctb.1999.1945
| |
| | issue = 2
| |
| | journal = [[Journal of Combinatorial Theory]] | series = Series B
| |
| | mr = 1750898
| |
| | pages = 243–273
| |
| | title = Oriented Hamiltonian paths in tournaments: a proof of Rosenfeld's conjecture
| |
| | volume = 78
| |
| | year = 2000a}}.
| |
| *{{citation
| |
| | last1 = Havet | first1 = Frédéric
| |
| | last2 = Thomassé | first2 = Stéphan
| |
| | doi = 10.1002/1097-0118(200012)35:4<244::AID-JGT2>3.0.CO;2-H
| |
| | issue = 4
| |
| | journal = Journal of Graph Theory
| |
| | mr = 1791347
| |
| | pages = 244–256
| |
| | title = Median orders of tournaments: a tool for the second neighborhood problem and Sumner's conjecture
| |
| | volume = 35
| |
| | year = 2000b}}.
| |
| *{{citation
| |
| | last1 = Kühn | first1 = Daniela | author1-link = Daniela Kühn
| |
| | last2 = Mycroft | first2 = Richard
| |
| | last3 = Osthus | first3 = Deryk
| |
| | doi = 10.1016/j.jctb.2010.12.006
| |
| | issue = 6
| |
| | journal = [[Journal of Combinatorial Theory]] | series = Series B
| |
| | mr = 2832810 | zbl=1234.05115
| |
| | pages = 415–447
| |
| | title = An approximate version of Sumner's universal tournament conjecture
| |
| | volume = 101
| |
| | year = 2011a}}.
| |
| *{{citation
| |
| | last1 = Kühn | first1 = Daniela | author1-link = Daniela Kühn
| |
| | last2 = Mycroft | first2 = Richard
| |
| | last3 = Osthus | first3 = Deryk
| |
| | arxiv = 1010.4430
| |
| | doi = 10.1112/plms/pdq035
| |
| | issue = 4
| |
| | journal = Proceedings of the London Mathematical Society | series = Third Series
| |
| | mr = 2793448 | zbl=1218.05034
| |
| | pages = 731–766
| |
| | title = A proof of Sumner's universal tournament conjecture for large tournaments
| |
| | volume = 102
| |
| | year = 2011b}}.
| |
| *{{citation
| |
| | last1 = Reid | first1 = K. B.
| |
| | last2 = Wormald | first2 = N. C.
| |
| | issue = 2-4
| |
| | journal = Studia Scientiarum Mathematicarum Hungarica
| |
| | mr = 787942
| |
| | pages = 377–387
| |
| | title = Embedding oriented ''n''-trees in tournaments
| |
| | volume = 18
| |
| | year = 1983}}.
| |
| *{{citation
| |
| | last = Rosenfeld | first = M.
| |
| | journal = [[Journal of Combinatorial Theory]] | series = Series B
| |
| | mr = 0285452
| |
| | pages = 93–99
| |
| | title = Antidirected Hamiltonian paths in tournaments
| |
| | volume = 12
| |
| | year = 1972}}.
| |
| *{{citation
| |
| | last = Thomason | first = Andrew
| |
| | doi = 10.2307/2000567
| |
| | issue = 1
| |
| | journal = Transactions of the American Mathematical Society
| |
| | mr = 837805
| |
| | pages = 167–180
| |
| | title = Paths and cycles in tournaments
| |
| | volume = 296
| |
| | year = 1986}}.
| |
| *{{citation
| |
| | last = Wormald | first = Nicholas C.
| |
| | contribution = Subtrees of large tournaments
| |
| | doi = 10.1007/BFb0071535
| |
| | location = Berlin
| |
| | mr = 731598
| |
| | pages = 417–419
| |
| | publisher = Springer
| |
| | series = Lecture Notes in Math.
| |
| | title = Combinatorial mathematics, X (Adelaide, 1982)
| |
| | volume = 1036
| |
| | year = 1983}}.
| |
| | |
| ==External links==
| |
| *[http://www.math.uiuc.edu/~west/openp/univtourn.html Sumner's Universal Tournament Conjecture (1971)], D. B. West, updated July 2008.
| |
| | |
| [[Category:Graph theory]]
| |
| [[Category:Conjectures]]
| |
Be sure that this prop is the exact size needed to carry one finish of the drywall towards the ceiling joists. The upright needs to be about 2 1/4 inches shorter than the space from the floor to ceiling joists. The cross piece needs to be about 4 ft long. Use the prop to help the opposite finish of the drywall when you fasten your end to the ceiling. Then transfer to the opposite finish and sink some screws into the drywall. Once each ends are secured, you may end screwing off the piece.
For functions of this text, let’s outline what I mean by pocket knife. Clearly it's a knife that is meant to be carried in your pocket. More particularly nevertheless, I am defining a pocket knife as a folding knife and as such I will be neglecting fixed blade knives and multitools, as they're outdoors the scope of this text. At 7.eight ounces, the ZT 0200 is a beast. I recommend this knife for a survival folder or for normal use round the house. The scale makes is cumbersome to hold every single day, to not mention the blade length makes it unlawful to carry in many locales.
The Clip Level blade is one of the most popular styles. The top facet of this knife is straight from the deal with to the mid blade area. It then curves downward and barely again as much as the tip of the knife The curved space on high of the innovative is known as the "clip". The sort of knife is widespread in mounted blade knives Wear gloves. Consider it this fashion, for those who lower your hands or fingers, you are done. You won't have the ability to whittle in any respect till it heals. Wear them, a minimum of until you have developed some actual talent at handling the knife
This is one other manifestation of the idiotic zero-tolerance policies adopted by colleges. On one hand, we've a pupil who made a mistake and brought it to the varsity’s attention voluntarily, and on the other hand, we have now a hypothetical of a student who maliciously brings a knife to highschool but will Spyderco Delica 4 Vs Paramilitary 2 (this article) get caught. Though the two conditions are each objectively and subjectively completely different, the college treats both the same — in order that they don’t need to train any widespread sense. It’s an absurdity, and yet one more information level demonstrating the decline of adult management in public training.
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