Landau's inequalities for tournament scores and a short proof of a theorem on transitive sub‐tournaments

Abstract

AbstractAo and Hanson, and Guiduli, Gyárfás, Thomassé and Weidl independently, proved the following result: For any tournament score sequence S = (s1, s2, … ,sn) with s1≤s2 ≤ … ≤ sn, there exists a tournament T on vertex set {1,2, …, n} such that the score of each vertex i is si and the sub‐tournaments of T on 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 inequalities which are individually stronger than the well‐known set of inequalities of Landau, but collectively the two sets of inequalities are equivalent. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 244–254, 2001