On cyclotomic nearly-doubly-regular tournaments

TL;DR

This paper constructs new cyclotomic nearly-doubly-regular tournaments, determines their spectrum, and links them to almost difference sets, providing evidence for the infinite existence of such tournaments under a number theory conjecture.

Contribution

It introduces a novel construction of cyclotomic nearly-doubly-regular tournaments and establishes their spectral properties through a new connection with combinatorial design theory.

Findings

Constructed new cyclotomic nearly-doubly-regular tournaments.

Determined the spectrum of these tournaments.

Provided conditional proof of infinite such tournaments assuming Hardy-Littlewood conjecture F.

Abstract

Nearly-doubly-regular tournaments have played significant roles in extremal graph theory. In this note, we construct new cyclotomic nearly-doubly-regular tournaments and determine their spectrum by establishing a new connection between cyclotomic nearly-doubly-regular tournaments and almost difference sets from combinatorial design theory. Furthermore, under the celebrated Hardy-Littlewood conjecture F in analytic number theory, our results confirm the conjecture due to Sergey Savchenko (J. Graph Theory {\bf 83} (2016), 44--77) on the existence of infinitely many nearly-doubly-regular tournaments with the canonical spectrum.