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arXiv · 2601.12250

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

Chi Hoi Yip·Semin Yoo·Shikang Yu
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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 $p\equiv3\pmod4$, 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.

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Chi Hoi Yip, Semin Yoo, Shikang Yu. 2026-01-18. Paley-type matrices and $1$-factorizations of complete graphs. https://arxiv.org/abs/2601.12250

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