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 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 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 . Transitivity 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. Equivalent conditions 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. Ramsey theory Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs. In particular,