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

Abstract

Ball, Ortega--Moreno, and Prodromou asked two questions about 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. These problems are motivated by two natural compatibility conditions between $1$-factorizations and the sign patterns of certain Paley-type matrices. Recently, Afifurrahman et al. made some partial progress on the second problem. In this paper, we completely resolve both problems. We prove that the first problem has a solution precisely when $p\equiv 3(\mathrm{mod}4)$, while the second problem has a solution for every odd prime $p$. We also solve a further problem of Ball et al. for cyclic groups of odd order, and more generally for all finite abelian groups of odd order.

arXiv comment16 pages, this version additionally resolves another open problem in Ball, Ortega--Moreno, and Prodromou