Dual Arc A Number Of Issues Of The Tournament

On Some Problems Of Double-arc TournamentsAbstractA competition where n participants play each other just once and where the ties are allowed, can be interpreted as a double-arc tournament a directed graph where each pair of vertices are joined by two arcs The score-vector of a double-arc tournament is defined as the vector of out degrees of vertices. In this paper we study the properties of the double-arc tournaments and give the numbers of some classes of score-vectors. The generating function for the cardinalities of the class of the double-arc touniaments is establislrcL Our study shows that the number of the double-arc tournaments has a closed relation to the partial order of their score-vectors. In order to reveal the relationship among tournaments, the cycle interchange and the interchange graph are introduced and studied. We show that the interchange graph is connected, and the connectivity of interchange graph is no less than a half of its diameter. TIE problem about ranking participants is also discussed.