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<p>Let <i>p</i> be a prime. Ball, Ortega-Moreno, and Prodromou asked about constructions of a 1-factor of a labelled <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2025_772_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(K^{p+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>K</mi> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> graph that is connected to a Hadamard-based 1-factorization of complete graphs. In this paper, we give some constructions on this problem, based on the primitive roots of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2025_772_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {F}_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>. In particular, we completely answered the problem for the case <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2025_772_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\equiv 5\pmod {8}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>5</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On constructing 1-factors of labelled complete graph

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月9日

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  • Muhammad Afifurrahman,
  • Devi Imulia Dian Primaskun,
  • Pritta Etriana Putri,
  • Laurence P. Wijaya

摘要

Let p be a prime. Ball, Ortega-Moreno, and Prodromou asked about constructions of a 1-factor of a labelled \(K^{p+1}\) K p + 1 graph that is connected to a Hadamard-based 1-factorization of complete graphs. In this paper, we give some constructions on this problem, based on the primitive roots of \(\mathbb {F}_p\) F p . In particular, we completely answered the problem for the case \(p\equiv 5\pmod {8}\) p ≡ 5 ( mod 8 ) .