Landau's inequalities for tournament scores and a short proof of a theoremon transitive sub-tournaments

Ao and Hanson, and Guiduli, Gyarfas, Thomasse and Weidl independently, prov ed the following result: For any tournament score sequence S = (s(1), s(2), ...,s(n)) with s(1) less than or equal to s(2) less than or equal to ... le ss than or equal to s(n), there exists a tournament Ton vertex set {1, 2,.. ., n} such that the score of each vertex i is si and the sub-tournaments of Ton both the even and the odd indexed vertices are transitive in the given order; that is, i dominates j whenever i >j and i =j (mod 2). In this note , we give a much shorter proof of the result. In the course of doing so, we show that the score sequence of a tournament satisfies a set of inequaliti es which are individually stronger than the well-known set of inequalities of Landau, but collectively the two sets of inequalities are equivalent. (C ) 2001 John Wiley & Sons, Inc.