A complete oriented graph (Skiena 1990, p. 175), i.e., a graph in which every pair of nodes is connected by a single uniquely directed edge. The first and second 3-node tournaments shown above are called a transitive triple and cyclic triple, respectively (Harary 1994, p. 204).
Tournaments (also called tournament graphs) are so named because an 
-node tournament graph correspond 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 set 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 
points for a loss.) The score sequence
for a given tournament is obtained from the set of outdegrees
sorted in nondecreasing order.
The number 
of nonisomorphic tournaments on 
, 3, 4, ... nodes are 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 symmetric group 
, define
 |
(1)
|
where
 |
(2)
|
with 
the number of group elements in
the conjugacy class of 
in 
, and 
is the number of 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)
|
(Davis 1954).
Every tournament contains an odd number of Hamiltonian paths (Rédei 1934; Szele 1943; Skiena 1990, p. 175). However, a tournament has a directed Hamiltonian cycle iff it is strongly connected (Foulkes 1960; Harary and Moser 1966; Skiena 1990, p. 175).
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).
In general, to fairly determine the best two players from 
contestants, 
rounds are required (Steinhaus 1999, p. 55).
