Tournament (graph theory)
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A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph in which every pair of distinct vertices is connected by a directed edge (often, called an arc) with any one of the two possible orientations.
Many of the important properties of tournaments were first investigated by H. G. Landau in Landau (1953) to model dominance relations in flocks of chickens. Current applications of tournaments include the study of voting theory and social choice theory among other things.
The name tournament originates from such a graph's interpretation as the outcome of a round-robin tournament in which every player encounters every other player exactly once, and in which no draws occur. In the tournament digraph, the vertices correspond to the players. The edge between each pair of players is oriented from the winner to the loser. If player beats player , then it is said that dominates . If every player beats the same number of other players (indegree = outdegree), the tournament is called regular.
Paths and cycles
This is easily shown by induction on : suppose that the statement holds for , and consider any tournament on vertices. Choose a vertex of and consider a directed path in . There is some such that . (One possibility is to let be maximal such that for every . Alternatively, let be minimal such that .)
is a directed path as desired. This argument also gives an algorithm for finding the Hamiltonian path. More efficient algorithms, that require examining only 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: for each vertex , and each in the range from three to the number of vertices in the tournament, there is a cycle of length containing . A tournament is -strongly connected if for every set of vertices of , is strongly connected. If the tournament is 4‑strongly connected, then each pair of vertices can be connected with a Hamiltonian path. For every set of at most arcs of a -strongly connected tournament , we have that has a Hamiltonian cycle. This result was extended by Bang-Jensen, Gutin & Yeo (1997).
A tournament in which and 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.
The following statements are equivalent for a tournament on vertices:
- is transitive.
- is a strict total ordering.
- is acyclic.
- does not contain a cycle of length 3.
- The score sequence (set of outdegrees) of is .
- has exactly one Hamiltonian path.
Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs. In particular, every tournament on vertices contains a transitive subtournament on vertices. The proof is simple: choose any one vertex to be part of this subtournament, and form the rest of the subtournament recursively on either the set of incoming neighbors of or the set of outgoing neighbors of , 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 .
Erdős & Moser (1964) proved that there are tournaments on vertices without a transitive subtournament of size Their proof uses a counting argument: the number of ways that a -element transitive tournament can occur as a subtournament of a larger tournament on labeled vertices is
and when is larger than , this number is too small to allow for an occurrence of a transitive tournament within each of the different tournaments on the same set of labeled vertices.
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 is called -paradoxical if for every -element subset of there is a vertex in such that for all . By means of the probabilistic method, Paul Erdős showed that for any fixed value of , if , then almost every tournament on is -paradoxical. On the other hand, an easy argument shows that any -paradoxical tournament must have at least players, which was improved to by Esther and George Szekeres (1965). There is an explicit construction of -paradoxical tournaments with players by Graham and Spencer (1971) namely the Paley tournament.
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) A nondecreasing sequence of integers is a score sequence if and only if :
1, 1, 1, 2, 4, 9, 22, 59, 167, 490, 1486, 4639, 14805, 48107, ...
Winston and Kleitman proved that for sufficiently large n:
where Takács later showed, using some reasonable but unproven assumptions, that
Together these provide evidence that:
Here 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. Let be a finite set of alternatives, and consider a list of linear orders over . We interpret each order as the preference ranking of a voter . The (strict) majority relation of over is then defined so that if and only if a majority of the voters prefer to , that is . If the number of voters is odd, then the majority relation forms the dominance relation of a tournament on vertex set .
By a lemma of McGarvey, every tournament on vertices can be obtained as the majority relation of at most voters. Results by Stearns and Erdős & Moser later established that voters are needed to induce every tournament on 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.
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