A tournament is a finite complete graph of positive graph order with exactly one direction assigned to each graph edge (Skiena 1990, p. 175), i.e., a directed graph in which every pair of distinct vertices is connected by a single directed edge. The first and second tournaments on three vertices shown above are called a transitive triple and cyclic triple, respectively (Harary 1994, p. 204).
Tournaments (also called tournament graphs) are so named because a tournament of graph order 
corresponds to a tournament in which each member of a group
of 
players plays all other 
players, and each game results in a win for one player and
a loss for the other. A so-called score sequence
can be associated with every tournament giving the multiset
of scores that would be obtained by the players in the tournament, with each win
counting as one point and each loss counting as no points. (A different scoring system
is used to compute a tournament's so-called tournament
matrix, with 1 point awarded for a win and 
point for a loss.) The score
sequence for a given tournament is obtained from the multiset
of outdegrees sorted in nondecreasing order.
The number 
of nonisomorphic tournaments on 
, 3, 4, ... vertices is 1,
2, 4, 12, 56, 456, ... (OEIS A000568; Moon
1968, Goldberg and Moon 1970; Harary and Palmer 1973, pp. 126 and 245; Reid
and Beineke 1978). Davis (1954) and Harary (1957) obtained a formula for these numbers
as a function of 
using the Pólya enumeration theorem.
For a permutation 
in the symmetric group 
, define
 |
(1)
|
where
 |
(2)
|
with 
the number of group
elements in the conjugacy class of 
in 
,
and 
is the number of permutation
cycles of length 
in the disjoint-cycle representation of any member of the class. Define
 |
(3)
|
where 
is the greatest common divisor of 
and 
. Then
 |
(4)
|
where the sum is over one representative of each conjugacy class of 
(Davis 1954, pp. 134-135).
Every tournament on at least two vertices contains an odd number of Hamiltonian paths (Rédei 1934, Szele 1943; Skiena 1990, p. 175). However, a tournament on at least two vertices has a directed Hamiltonian cycle iff it is strongly connected (Foulkes 1960, Harary and Moser 1966; Skiena 1990, p. 175).
Divoux and Wang (2026) determined the asymptotic maximum number 
of directed Hamiltonian
cycles in an 
-vertex
tournament,
 |
(5)
|
as 
. They also proved that the
maximum number 
of directed Hamiltonian paths satisfies
 |
(6)
|
The counts identify cycles differing only in their starting graph vertex. Divoux and Wang (2026) credit GPT-5.6 with finding the lower-bound argument and a matching upper-bound argument. The authors revised and simplified the proofs.
The term "tournament" also refers to an arrangement by which teams or players play against certain other teams or players in order to determine who is the best.
In a "cup" tournament of 
teams, teams play pairwise in a sequence of 
-finals, ..., 1/8-finals, quarterfinals, semifinals,
and finals, with winners from each round playing other winners in the next round
and losers being eliminated at each round. The second-place prize is usually awarded
to the team that loses in the finals. However, this practice is unfair since the
second-place team has not been required to play against the teams that were eliminated
by the first-place (and presumably best) team, and therefore might actually be worse
than one of the teams eliminated earlier by the best team (Steinhaus 1999).
If 
contestants have a fixed strict
total order and every game is won by the better player,
the minimum worst-case number of individual games needed to determine the best two
is ![n+[log_2n]-2](/images/equations/Tournament/Inline26.svg)
. Here ![[x]](/images/equations/Tournament/Inline27.svg)
is the ceiling function.
These games may be organized into fewer parallel rounds.
