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 ${\displaystyle v}$ , and each ${\displaystyle k}$  in the range from three to the number of vertices in the tournament, there is a cycle of length ${\displaystyle k}$  containing ${\displaystyle v}$ .^[4] A tournament ${\displaystyle T}$ is ${\displaystyle k}$ -strongly connected if for every set ${\displaystyle U}$  of ${\displaystyle k-1}$  vertices of ${\displaystyle T}$ , ${\displaystyle T-U}$  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 ${\displaystyle B}$  of at most ${\displaystyle k-1}$  arcs of a ${\displaystyle k}$ -strongly connected tournament ${\displaystyle T}$ , we have that ${\displaystyle T-B}$  has a Hamiltonian cycle.^[6] This result was extended by Bang-Jensen, Gutin & Yeo (1997).

A tournament in which ${\displaystyle ((a\rightarrow b)}$  and ${\displaystyle (b\rightarrow c))}$  ${\displaystyle \Rightarrow }$  ${\displaystyle (a\rightarrow c)}$  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 edit

The following statements are equivalent for a tournament ${\displaystyle T}$  on ${\displaystyle n}$  vertices:

1. ${\displaystyle T}$  is transitive.
2. ${\displaystyle T}$  is a strict total ordering.
3. ${\displaystyle T}$  is acyclic.
4. ${\displaystyle T}$  does not contain a cycle of length 3.
5. The score sequence (set of outdegrees) of ${\displaystyle T}$  is ${\displaystyle \{0,1,2,\ldots ,n-1\}}$ .
6. ${\displaystyle T}$  has exactly one Hamiltonian path.

Ramsey theory edit

Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs. In particular, every tournament on ${\displaystyle n}$  vertices contains a transitive subtournament on ${\displaystyle 1+\lfloor \log _{2}n\rfloor }$  vertices.^[7] The proof is simple: choose any one vertex ${\displaystyle v}$  to be part of this subtournament, and form the rest of the subtournament recursively on either the set of incoming neighbors of ${\displaystyle v}$  or the set of outgoing neighbors of ${\displaystyle v}$ , 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 ${\displaystyle n}$ .

Erdős & Moser (1964) proved that there are tournaments on ${\displaystyle n}$  vertices without a transitive subtournament of size ${\displaystyle 2+2\lfloor \log _{2}n\rfloor }$  Their proof uses a counting argument: the number of ways that a ${\displaystyle k}$ -element transitive tournament can occur as a subtournament of a larger tournament on ${\displaystyle n}$  labeled vertices is

${\displaystyle {\binom {n}{k}}k!2^{{\binom {n}{2}}-{\binom {k}{2}}},}$ 

and when ${\displaystyle k}$  is larger than ${\displaystyle 2+2\lfloor \log _{2}n\rfloor }$ , this number is too small to allow for an occurrence of a transitive tournament within each of the ${\displaystyle 2^{\binom {n}{2}}}$  different tournaments on the same set of ${\displaystyle n}$  labeled vertices.

Paradoxical tournaments edit

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 ${\displaystyle T=(V,E)}$  is called ${\displaystyle k}$ -paradoxical if for every ${\displaystyle k}$ -element subset ${\displaystyle S}$  of ${\displaystyle V}$  there is a vertex ${\displaystyle v_{0}}$  in ${\displaystyle V\setminus S}$  such that ${\displaystyle v_{0}\rightarrow v}$  for all ${\displaystyle v\in S}$ . By means of the probabilistic method, Paul Erdős showed that for any fixed value of ${\displaystyle k}$ , if ${\displaystyle |V|\geq k^{2}2^{k}\ln(2+o(1))}$ , then almost every tournament on ${\displaystyle V}$  is ${\displaystyle k}$ -paradoxical.^[8] On the other hand, an easy argument shows that any ${\displaystyle k}$ -paradoxical tournament must have at least ${\displaystyle 2^{k+1}-1}$  players, which was improved to ${\displaystyle (k+2)2^{k-1}-1}$  by Esther and George Szekeres (1965).^[9] There is an explicit construction of ${\displaystyle k}$ -paradoxical tournaments with ${\displaystyle k^{2}4^{k-1}(1+o(1))}$  players by Graham and Spencer (1971) namely the Paley tournament.

Condensation edit

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]

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 ${\displaystyle (s_{1},s_{2},\ldots ,s_{n})}$  is a score sequence if and only if :

1. ${\displaystyle 0\leq s_{1}\leq s_{2}\leq \cdots \leq s_{n}}$ 
2. ${\displaystyle s_{1}+s_{2}+\cdots +s_{i}\geq {i \choose 2},{\text{ for }}i=1,2,\ldots ,n-1}$ 
3. ${\displaystyle s_{1}+s_{2}+\cdots +s_{n}={n \choose 2}.}$ 

Let ${\displaystyle s(n)}$  be the number of different score sequences of size ${\displaystyle n}$ . The sequence ${\displaystyle s(n)}$  (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:

${\displaystyle s(n)>c_{1}4^{n}n^{-5/2},}$ 

where ${\displaystyle c_{1}=0.049.}$  Takács later showed, using some reasonable but unproven assumptions, that

${\displaystyle s(n)<c_{2}4^{n}n^{-5/2},}$ 

where ${\displaystyle c_{2}<4.858.}$ 

Together these provide evidence that:

${\displaystyle s(n)\in \Theta (4^{n}n^{-5/2}).}$ 

Here ${\displaystyle \Theta }$  signifies an asymptotically tight bound.

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

In social choice theory, tournaments naturally arise as majority relations of preference profiles.^[11] Let ${\displaystyle A}$  be a finite set of alternatives, and consider a list ${\displaystyle P=(\succ _{1},\dots ,\succ _{n})}$  of linear orders over ${\displaystyle A}$ . We interpret each order ${\displaystyle \succ _{i}}$  as the preference ranking of a voter ${\displaystyle i}$ . The (strict) majority relation ${\displaystyle \succ _{\text{maj}}}$  of ${\displaystyle P}$  over ${\displaystyle A}$  is then defined so that ${\displaystyle a\succ _{\text{maj}}b}$  if and only if a majority of the voters prefer ${\displaystyle a}$  to ${\displaystyle b}$ , that is ${\displaystyle |\{i\in [n]:a\succ _{i}b\}|>|\{i\in [n]:b\succ _{i}a\}|}$ . If the number ${\displaystyle n}$  of voters is odd, then the majority relation forms the dominance relation of a tournament on vertex set ${\displaystyle A}$ .

By a lemma of McGarvey, every tournament on ${\displaystyle m}$  vertices can be obtained as the majority relation of at most ${\displaystyle m(m-1)}$  voters.^[11][12] Results by Stearns and Erdős & Moser later established that ${\displaystyle \Theta (m/\log m)}$  voters are needed to induce every tournament on ${\displaystyle m}$  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]

• Oriented graph
• Paley tournament
• Sumner's conjecture
• Tournament solution

• 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", Graphs and Combinatorics, 3: 239–250.
• Graham, R. L.; Spencer, J. H. (1971), "A constructive solution to a tournament problem", Canadian Mathematical Bulletin, 14: 45–48, doi:10.4153/cmb-1971-007-1, MR 0292715.
• Harary, Frank; Moser, Leo (1966), "The theory of round robin tournaments", American Mathematical Monthly, 73 (3): 231–246, doi:10.2307/2315334, JSTOR 2315334.
• Havet, Frédéric (2013), "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), "On subtournaments of a tournament", 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", Journal of Combinatorial Theory, 9 (3): 225–238, doi:10.1016/S0021-9800(70)80061-8.
• Szekeres, E.; Szekeres, G. (1965), "On a problem of Schütte and Erdős", 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, 23 (3), Applied Probability Trust: 557–585, doi:10.2307/1427622, JSTOR 1427622.
• Thomassen, Carsten (1980), "Hamiltonian-Connected Tournaments", Journal of Combinatorial Theory, Series B, 28 (2): 142–163, doi:10.1016/0095-8956(80)90061-1.
• Yao, T.X. (1989), "On Reid conjecture of score sets for tournaments", Chinese Sci. Bull., 34: 804–808.

This article incorporates material from tournament on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.