On a conjecture of Brualdi and Shen on block transitive tournaments

Abstract

Abstract:The following conjecture of Brualdi and Shen is proven in this paper: let n be partitioned into natural numbers no one of which is greater than ðn þ 1Þ=2. Then, given any sequence of wins for the players of some tournament among n players, there is a partition of the players into blocks with cardinalities given by those numbers, and a tournament with the given sequence of wins, that is transitive on the players within each block.

Transitive partitions in realizations of tournament score sequences

“…In particular, by generalizing the results of Guiduli et al on 2-partition transitive tournaments to tournaments with unequal 2-partitions, we will obtain as a corollary a very natural proof of a stronger version of Corollary 1.3 as suggested by Acosta et al [1] without the need for a separate proof of the 3-partition result.…”

“…The techniques used in [1] to prove Theorem 1.2 combine an inductive approach with an explicit determination of the 3-partition based on the score sequence of the tournament T. As a result, it remains possible to use this proof to construct a specific 3-partition transitive tournament using a recursive algorithm derived from the inductive step, although it is not clear how efficient such an algorithm will be.…”

“…Theorem 1.2 (Acosta et al [1]). Let T be a tournament, of order n with n = n 1 + n 2 +n 3 such that n /2 ≥ n 1 ≥ n 2 ≥ n 3 ≥ 1.…”

“…In addition to the proof of Theorem 1.1, Brualdi and Shen also conjectured the following result on 3-partition transitive tournaments, which was proven by Acosta, Bassa, Chaikin, Riehl, Tingstad, Zhao, and Kleitman. Theorem 1.2 (Acosta et al [1]). Let T be a tournament, of order n with…”

Reconstruction of complete interval tournaments. II

“…E.g. in the papers [8,16,18,19,20,21,26,30,32,34,36,45,68,84,85,88,90,98] the graphical sequences, while in the papers [1,2,3,7,8,11,17,27,28,29,31,33,37,49,48,50,55,58,57,60,61,62,64,65,66,69,78,79,82,94,86,87,97,100,…”