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<p>Let <i>m</i> be a positive integer. A group <i>G</i> is said to be an <i>m</i>-DCI-group or an <i>m</i>-CI-group if <i>G</i> has the <i>k</i>-DCI property or <i>k</i>-CI property for all positive integers <i>k</i> at most <i>m</i>, respectively. Let <i>G</i> be a dihedral group of order 2<i>n</i> with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1449_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Qu and Yu proved that <i>G</i> is an <i>m</i>-DCI-group or <i>m</i>-CI-group, for every <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1449_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">\(m\in \{1,2,3\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, if and only if <i>n</i> is odd. In this paper, it is shown that <i>G</i> is a 4-DCI-group if and only if <i>n</i> is odd and not divisible by 9, and <i>G</i> is a 4-CI-group if and only if <i>n</i> is odd.</p>

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On isomorphisms of tetravalent Cayley digraphs over dihedral groups

  • 期刊论文

  • 发表日期：

    2025年9月3日

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  • Jin-Hua Xie,
  • Zai Ping Lu,
  • Yan-Quan Feng

摘要

Let m be a positive integer. A group G is said to be an m-DCI-group or an m-CI-group if G has the k-DCI property or k-CI property for all positive integers k at most m, respectively. Let G be a dihedral group of order 2n with \(n\ge 3\) n ≥ 3 . Qu and Yu proved that G is an m-DCI-group or m-CI-group, for every \(m\in \{1,2,3\}\) m ∈ { 1 , 2 , 3 } , if and only if n is odd. In this paper, it is shown that G is a 4-DCI-group if and only if n is odd and not divisible by 9, and G is a 4-CI-group if and only if n is odd.