A constructive solution to a problem of ranking tournaments

TL;DR

This paper provides an explicit construction method for unrankable tournaments using skew Hadamard difference sets, addressing a longstanding problem in combinatorial design and tournament theory.

Contribution

It introduces a new explicit construction technique for unrankable tournaments based on skew Hadamard difference sets, expanding the known examples.

Findings

Constructed many unrankable tournaments explicitly

Connected combinatorial design theory with tournament ranking problems

Enhanced understanding of the structure of unrankable tournaments

Abstract

A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erd\H{o}s and J. W. Moon. By probabilistic methods, the existence of "unrankable" tournaments was proved. On the other hand, they also mentioned the problem of explicit constructions. However, there seems to be only a few of explicit constructions of such tournaments. In this note, we give a construction of many such tournaments by using skew Hadamard difference sets which have been investigated in combinatorial design theory.