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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==
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.
 
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>.
 
==Partial results==
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>
*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>
*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>
*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>
*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"/>
*Every <math>(2n-2)</math>-vertex tournament contains as a subgraph every <math>n</math>-vertex [[rooted tree]].<ref>{{harvtxt|Havet|Thomassé|2000b}}.</ref>
 
==Related conjectures==
{{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}}.
 
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.
 
{{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.
 
==Notes==
{{reflist|colwidth=30em}}
 
==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]]

Revision as of 23:33, 22 February 2014

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