Tournament
An oriented graph (cf. also Graph, oriented) without loops, each pair of vertices of which are joined by an arc in exactly one direction. A tournament with vertices can be regarded as the outcome of a competition with
players, the rules of which forbid draws. The notion of a tournament is used for ordering
objects by the method of pairwise comparison. In this connection it finds application in biology, sociology, etc.
A tournament is said to be transitive if its vertices can be indexed by the numbers in such a way that there is an arc going from
to
if and only if
. There are no circuits in a transitive tournament. A tournament is said to be strong if for any ordered pair of vertices
there exists a directed path from
to
. A set of arcs in a tournament is called compatible if there are no circuits in the subgraph formed by these arcs and the vertices incident to them. The maximum cardinality of a set of compatible arcs is a measure of compatibility in the definition of "the winner" of the tournament. Every tournament contains a subset of compatible arcs of cardinality not less than
. Another measure of compatibility is the ratio of the number of transitive
-vertex subtournaments of a tournament with
vertices to the number of strong
-vertex subtournaments. The maximum number of strong
-vertex subtournaments of a tournament with
vertices is equal to
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A tournament is strong if and only if it has a spanning cycle (Hamiltonian circuit). Every strong tournament with vertices has a circuit of length
for
. Every tournament has a spanning path (Hamiltonian path).
The outdegrees of a tournament with
vertices satisfy the equation
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Suppose that a set of integers satisfies the condition
. Then a tournament with outdegrees
exists if and only if for any
the inequality
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holds, with equality for . Furthermore, a tournament is strong if and only if
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If and
are two subtournaments of a tournament
and if there exists an arc
for each pair of vertices
in
and
in
, then one writes
. Suppose that the set of vertices of a tournament is partitioned into non-intersecting subsets
. Suppose further that either
or
for
. Then the partition defines an equivalence relation on the vertices of
. A tournament is called simple if no non-trivial equivalence relation can be defined on its vertices. Every tournament with
vertices is a subtournament of some simple tournament with
vertices. A tournament
with
vertices is a subtournament of some simple tournament with
vertices if and only if
is neither a circuit with three vertices nor a non-trivial transitive tournament with an odd number of vertices. The number of pairwise non-isomorphic tournaments with
vertices is asymptotically equal to
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The number of distinct tournaments with indexed vertices is equal to
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The generating functions and
for tournaments and strong tournaments, respectively, are related by the formula:
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Every tournament with vertices,
, that is not strong is uniquely recoverable from the family of its
-vertex subtournaments.
References
[1] | F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9 |
[2] | J.W. Moon, "Topics on tournaments" , Holt, Rinehart & Winston (1968) |
Comments
A random tournament over is defined by making random choices of an arc
or
for each pair of different vertices
, the choice being equiprobable and independent for all different pairs. Cf. [2] for many results on random tournaments.
References
[a1] | F. Harary, L. Moser, "The theory of round robin tournaments" Amer. Math. Monthly , 73 (1966) pp. 231–246 |
[a2] | L. Comtet, "Advanced combinatorics" , Reidel (1974) pp. 68ff (Translated from French) |
Tournament. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tournament&oldid=49004