Paley-type matrices and $1$-factorizations of complete graphs

TL;DR

The paper resolves the existence of a $1$-factor of $K_{p+1}$ compatible with the Paley-type sign pattern for all odd primes $p$. It combines an explicit construction in the $p ≡ 1$ (mod 8) case, relying on a primitive root $a$ and a four-part partition of the noncentral vertices to ensure edges join a quadratic residue with a non-residue and have distinct lengths, with a quartic residuosity condition on $a$. The existence of such an $a$ for large $p$ is established via Weil-bound estimates on character sums, while small primes are verified computationally, completing the proof for all odd primes. Consequently, translating the constructed $1$-factor through ${f F}_p$ yields a $1$-factorization of $K_{p+1}$ compatible with the sign pattern of the Paley-type matrix $H_p$, extending the Paley- Hadamard framework to all odd primes.

Abstract

Ball, Ortega--Moreno, and Prodromou asked whether, for every odd prime $p$, one can find a $1$-factor of the complete graph $K_{p+1}$ with some arithmetic restrictions related to quadratic residues. This problem is motivated by $1$-factorizations that are compatible with the sign pattern of certain Paley-type matrices. Recently, Afifurrahman et al. made some partial progress. In this paper, we completely resolve the problem.

Theorems & Definitions (16)

• Theorem 1.2
• Corollary 1.3
• Proposition 2.1
• Claim 2.2
• proof
• Claim 2.3
• proof
• Claim 2.4
• proof
• Lemma 3.1