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

TL;DR

This paper completely resolves a problem about finding specific 1-factors in complete graphs related to Paley-type matrices and quadratic residues, advancing understanding in combinatorial design and graph theory.

Contribution

It provides a complete solution to the existence problem of 1-factors in complete graphs with arithmetic restrictions linked to Paley-type matrices.

Findings

Established existence of 1-factors with quadratic residue conditions for all odd primes

Connected 1-factorizations to Paley-type matrix sign patterns

Resolved a previously open problem in combinatorial design theory

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.