Tournament (graph theory) on arvau

Wikipedia / Tournament (graph theory)

In graph theory, a tournament is a directed graph with exactly one edge between each two vertices, in one of the two possible directions. Equivalently, a tournament is an orientation of an undirected complete graph. (However, as directed graphs, tournaments are not complete: complete directed graphs have two edges, in both directions, between each two vertices.) Equivalently, a tournament is a complete asymmetric relation.

The name tournament comes from interpreting the graph as the outcome of a round-robin tournament, a game where each player is paired against every other exactly once. In a tournament, the vertices represent the players, and the edges between players point from the winner to the loser.

Many of the important properties of tournaments were investigated by H. G. Landau in 1953 to model dominance relations in flocks of chickens. Tournaments are also heavily studied in voting theory, where they can represent partial information about voter preferences among multiple candidates, and are central to the definition of Condorcet methods.

If every player beats the same number of other players (indegree − outdegree = 0) the tournament is called regular. The number of unlabeled regular tournaments with 2n+1 vertices goes:

1, 1, 1, 3, 15, 1223, 1495297, 18400989629, 2406183070160597,...

Paths and cycles

Any tournament on a finite of vertices contains a Hamiltonian path vertices (Rédei 1934).

This is shown by induction of the edges, are known. The Hamiltonian paths are in one-to-one correspondence with the minimal feedback arc sets of the tournament. 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.

Another basic result on tournaments is that every strongly connected tournament has a Hamiltonian cycle. More strongly, every strongly connected tournament is vertex pancyclic. is strongly connected. If the tournament is 4‑strongly connected, then each pair of vertices can be connected with a Hamiltonian path. has a Hamiltonian cycle. This result was extended by .

Transitivity

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

vertices:

is transitive.
is a strict total ordering.
is acyclic.
does not contain a cycle of length 3.
.
has exactly one Hamiltonian path.

Ramsey theory

Transitive tournaments play a role in Ramsey theory analogous to that of cliques, 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. However, showed that this bound is not tight for some larger values of.

Their proof uses a counting argument labeled vertices.

Paradoxical tournaments

. By means of the probabilistic method, Paul Erdős-paradoxical. by Esther and George Szekeres in 1965. 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.

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) is a score sequence if and only if:

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:

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

Together these provide evidence that:

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. of linear orders as the preference ranking.

voters. Results by Stearns vertices.

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.