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<p>In this paper, we show that if <i>G</i> is 3-connected, then the Cartesian product of graphs <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/373_2026_3044_IEq1_HTML.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="120" Type="Linedraw" Width="49" /> </InlineMediaObject> </InlineEquation> embeds on the torus if and only if <i>G</i> is outer-cylindrical and <i>H</i> is a path on two vertices, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(P_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. As a by-product of our work, we also show that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/373_2026_3044_IEq3_HTML.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="120" Type="Linedraw" Width="60" /> </InlineMediaObject> </InlineEquation> has genus two.</p>

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Toroidal Cartesian Products Where One Factor is 3-Connected

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月16日

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  • Elizabeth Badgett,
  • Christian Millichap,
  • Kenta Noguchi

摘要

In this paper, we show that if G is 3-connected, then the Cartesian product of graphs embeds on the torus if and only if G is outer-cylindrical and H is a path on two vertices, \(P_2\) P 2 . As a by-product of our work, we also show that has genus two.