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 $p$.
From: Chi Hoi Yip [view email]
[v1]
Sun, 18 Jan 2026 04:01:40 UTC (10 KB)
[v2]
Thu, 6 Aug 2026 06:39:01 UTC (12 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。