# Tournament (graph theory) Information

*https://en.wikipedia.org/wiki/Tournament_graph*

Tournament | |
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Vertices | |

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Table of graphs and parameters |

A **tournament** is a
directed graph (digraph) obtained by assigning a direction for each edge in an
undirected
complete graph. That is, it is an
orientation of a complete graph, or equivalently a directed graph in which every pair of distinct
vertices is connected by a directed edge (often, called an **arc**) with any one of the two possible orientations.

Many of the important properties of tournaments were first investigated by H. G. Landau in Landau (1953) to model dominance relations in flocks of chickens. Current applications of tournaments include the study of voting theory and social choice theory among other things.

The name *tournament* originates from such a graph's interpretation as the outcome of a
round-robin tournament in which every player encounters every other player exactly once, and in which no draws occur. In the tournament digraph, the vertices correspond to the players. The edge between each pair of players is oriented from the winner to the loser. If player beats player , then it is said that *dominates* . If every player beats the same number of other players (*indegree* = *outdegree*), the tournament is called *regular*.

## Paths and cycles

**Theorem** — Any tournament on a
finite number of vertices contains a
Hamiltonian path, i.e., directed path on all vertices (
Rédei 1934).

This is easily shown by induction on : suppose that the statement holds for , and consider any tournament on vertices. Choose a vertex of and consider a directed path in . There is some such that . (One possibility is to let be maximal such that for every . Alternatively, let be minimal such that .)

is a directed path as desired. This argument also gives an algorithm for finding the Hamiltonian path. More efficient algorithms, that require examining only of the edges, are known. The Hamiltonian paths are in one-to-one correspondence with the minimal
feedback arc sets of the tournament.^{
[1]} Rédei's theorem is the special case for complete graphs of the
Gallai–Hasse–Roy–Vitaver theorem, relating the lengths of paths in orientations of graphs to the
chromatic number of these graphs.^{
[2]}

Another basic result on tournaments is that every
strongly connected tournament has a
Hamiltonian cycle.^{
[3]} More strongly, every strongly connected tournament is
vertex pancyclic: for each vertex , and each in the range from three to the number of vertices in the tournament, there is a cycle of length containing .^{
[4]} A tournament is -strongly connected if for every set of vertices of , is strongly connected.
If the tournament is 4‑strongly connected, then each pair of vertices can be connected with a Hamiltonian path.^{
[5]} For every set of at most arcs of a -strongly connected tournament , we have that has a Hamiltonian cycle.^{
[6]} This result was extended by
Bang-Jensen, Gutin & Yeo (1997).

## Transitivity

A tournament in which and is called **transitive**. In other words, in a transitive tournament, the vertices may be (strictly)
totally ordered by the edge relation, and the edge relation is the same as
reachability.

### Equivalent conditions

The following statements are equivalent for a tournament on vertices:

- is transitive.
- is a strict total ordering.
- is acyclic.
- does not contain a cycle of length 3.
- The score sequence (set of outdegrees) of is .
- has exactly one Hamiltonian path.

### Ramsey theory

Transitive tournaments play a role in
Ramsey theory analogous to that of
cliques in undirected graphs. In particular, every tournament on vertices contains a transitive subtournament on vertices.^{
[7]} The proof is simple: choose any one vertex to be part of this subtournament, and form the rest of the subtournament recursively on either the set of incoming neighbors of or the set of outgoing neighbors of , whichever is larger. For instance, every tournament on seven vertices contains a three-vertex transitive subtournament; the
Paley tournament on seven vertices shows that this is the most that can be guaranteed (
Erdős & Moser 1964). However,
Reid & Parker (1970) showed that this bound is not tight for some larger values of .

Erdős & Moser (1964) proved that there are tournaments on vertices without a transitive subtournament of size Their proof uses a counting argument: the number of ways that a -element transitive tournament can occur as a subtournament of a larger tournament on labeled vertices is

and when is larger than , this number is too small to allow for an occurrence of a transitive tournament within each of the different tournaments on the same set of labeled vertices.

### Paradoxical tournaments

A player who wins all games would naturally be the tournament's winner. However, as the existence of non-transitive tournaments shows, there may not be such a player. A tournament for which every player loses at least one game is called a 1-paradoxical tournament. More generally, a tournament is called -paradoxical if for every -element subset of there is a vertex in such that for all . By means of the
probabilistic method,
Paul Erdős showed that for any fixed value of , if , then almost every tournament on is -paradoxical.^{
[8]} On the other hand, an easy argument shows that any -paradoxical tournament must have at least players, which was improved to by
Esther and
George Szekeres (1965).^{
[9]} There is an explicit construction of -paradoxical tournaments with players by
Graham and Spencer (1971) namely the
Paley tournament.

### Condensation

The
condensation of any tournament is itself a transitive tournament. Thus, even for tournaments that are not transitive, the strongly connected components of the tournament may be totally ordered.^{
[10]}

## Score sequences and score sets

The score sequence of a tournament is the nondecreasing sequence of outdegrees of the vertices of a tournament. The score set of a tournament is the set of integers that are the outdegrees of vertices in that tournament.

**Landau's Theorem (1953)** A nondecreasing sequence of integers is a score sequence if and only if :

Let be the number of different score sequences of size . The sequence (sequence A000571 in the OEIS) starts as:

1, 1, 1, 2, 4, 9, 22, 59, 167, 490, 1486, 4639, 14805, 48107, ...

Winston and Kleitman proved that for sufficiently large *n*:

where Takács later showed, using some reasonable but unproven assumptions, that

where

Together these provide evidence that:

Here signifies an asymptotically tight bound.

Yao showed that every nonempty set of nonnegative integers is the score set for some tournament.

## Majority relations

In
social choice theory, tournaments naturally arise as majority relations of preference profiles.^{
[11]}
Let be a finite set of alternatives, and consider a list of
linear orders over . We interpret each order as the
preference ranking of a voter . The (strict) majority relation of over is then defined so that if and only if a majority of the voters prefer to , that is . If the number of voters is odd, then the majority relation forms the dominance relation of a tournament on vertex set .

By a lemma of McGarvey, every tournament on vertices can be obtained as the majority relation of at most voters.^{
[11]}^{
[12]} Results by
Stearns and Erdős & Moser later established that voters are needed to induce every tournament on vertices.^{
[13]}

Laslier (1997) studies in what sense a set of vertices can be called the set of "winners" of a tournament. This revealed to be useful in Political Science to study, in formal models of political economy, what can be the outcome of a democratic process.^{
[14]}

## See also

## Notes

**^**Bar-Noy & Naor (1990).**^**Havet (2013).**^**Camion (1959).**^**Moon (1966), Theorem 1.**^**Thomassen (1980).**^**Fraisse & Thomassen (1987).**^**Erdős & Moser (1964).**^**Erdős (1963)**^**Szekeres, E.; Szekeres, G. (1965). "On a problem of Schütte and Erdős".*Math. Gaz*.**49**: 290–293.**^**Harary & Moser (1966), Corollary 5b.- ^
^{a}^{b}Brandt, Felix and Brill, Markus and Harrenstein, Paul, "Tournament Solutions". Chapter 3 in: Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016).*Handbook of Computational Social Choice*. Cambridge University Press. ISBN 9781107060432. ( free online version) **^**McGarvey, David C. (1953). "A Theorem on the Construction of Voting Paradoxes".*Econometrica*.**21**(4): 608–610. doi: 10.2307/1907926. JSTOR 1907926.**^**Stearns, Richard (1959). "The Voting Problem".*The American Mathematical Monthly*.**66**(9): 761–763. doi: 10.2307/2310461. JSTOR 2310461.**^**Austen-Smith, D. and J. Banks, Positive Political theory, 1999, The University of Michigan Press.

## References

- Bang-Jensen, J.;
Gutin, G.; Yeo, A. (1997), "Hamiltonian Cycles Avoiding Prescribed Arcs in Tournaments",
*Combinatorics, Probability and Computing*,**6**: 255–261. - Bar-Noy, A.;
Naor, J. (1990), "Sorting, Minimal Feedback Sets and Hamilton Paths in Tournaments",
*SIAM Journal on Discrete Mathematics*,**3**(1): 7–20, doi: 10.1137/0403002. - Camion, Paul (1959),
"Chemins et circuits hamiltoniens des graphes complets",
*Comptes Rendus de l'Académie des Sciences de Paris*(in French),**249**: 2151–2152. -
Erdős, P. (1963),
"On a problem in graph theory" (PDF),
*The Mathematical Gazette*,**47**: 220–223, JSTOR 3613396, MR 0159319. -
Erdős, P.;
Moser, L. (1964),
"On the representation of directed graphs as unions of orderings" (PDF),
*Magyar Tud. Akad. Mat. Kutató Int. Közl.*,**9**: 125–132, MR 0168494. - Fraisse, P.; Thomassen, C. (1987), "A constructive solution to a tournament problem",
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Harary, Frank;
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"Section 3.1: Gallai–Roy Theorem and related results" (PDF),
*Orientations and colouring of graphs*, Lecture notes for the summer school SGT 2013 in Oléron, France, pp. 15–19 - Landau, H.G. (1953), "On dominance relations and the structure of animal societies. III. The condition for a score structure",
*Bulletin of Mathematical Biophysics*,**15**(2): 143–148, doi: 10.1007/BF02476378. - Laslier, J.-F. (1997),
*Tournament Solutions and Majority Voting*, Springer. - Moon, J. W. (1966),
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*Canadian Mathematical Bulletin*,**9**(3): 297–301, doi: 10.4153/CMB-1966-038-7. -
Rédei, László (1934), "Ein kombinatorischer Satz",
*Acta Litteraria Szeged*,**7**: 39–43. - Reid, K.B.; Parker, E.T. (1970), "Disproof of a conjecture of Erdös and Moser",
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*The Mathematical Gazette*,**49**: 290–293, doi: 10.2307/3612854, MR 0186566. -
Takács, Lajos (1991), "A Bernoulli Excursion and Its Various Applications",
*Advances in Applied Probability*, Applied Probability Trust,**23**(3): 557–585, doi: 10.2307/1427622, JSTOR 1427622. -
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