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On cyclotomic nearly-doubly-regular tournaments

Shohei Satake

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.

On cyclotomic nearly-doubly-regular tournaments

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.

Paper Structure

This paper contains 4 sections, 7 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem-main1']}
  3. Proof of Theorem \ref{['theorem-main2']}
  4. Concluding remarks

Key Result

Theorem 1

Let $q\equiv 5 \pmod{8}$ be a prime power (equivalently, $\ell=2$). Then $CT_q$ is nearly-doubly-regular if and only if $q=s^2+4$ for some odd integer $s$.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Conjecture 3: The Hardy-Littlewood conjecture F, HL23
  • Definition 4: Almost difference set
  • Definition 5: Cayley digraph
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Theorem 8: DHL99, N14
  • ...and 11 more