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<p>In 2009, Chmutov introduced the concept of partial-duality for a ribbon graph <i>G</i>. In 2020, Gross, Mansour and Tucker systematically enumerated all possible partial-duals of <i>G</i> by genus and also introduced the notion of the partial-dual genus polynomial for a ribbon graph <i>G</i>. In this paper, we investigate many aspects of the partial-duality of plane ribbon graphs. Firstly, we derive a formula for the maximum partial-dual genus of any plane ribbon graph. Secondly, we establish a recurrence relation for the partial-dual genus polynomial of plane ribbon graphs <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1465_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(G-e\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>-</mo> <mi>e</mi> </mrow> </math></EquationSource> </InlineEquation> and <i>G</i>. These recurrence relations provide a new approach to calculate the partial-dual genus polynomials for some plane ribbon graphs. Additionally, we establish the asymptotic normality of the partial-dual genus distributions for specific plane ribbon graphs. Furthermore, we employ a plane ribbon graph to construct a minimal counterexample to the interpolation conjecture.</p>

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Partial-duals for plane ribbon graphs

  • 期刊论文

  • 发表日期：

    2025年9月7日

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  • Qiyao Chen,
  • Yichao Chen

摘要

In 2009, Chmutov introduced the concept of partial-duality for a ribbon graph G. In 2020, Gross, Mansour and Tucker systematically enumerated all possible partial-duals of G by genus and also introduced the notion of the partial-dual genus polynomial for a ribbon graph G. In this paper, we investigate many aspects of the partial-duality of plane ribbon graphs. Firstly, we derive a formula for the maximum partial-dual genus of any plane ribbon graph. Secondly, we establish a recurrence relation for the partial-dual genus polynomial of plane ribbon graphs \(G-e\) G - e and G. These recurrence relations provide a new approach to calculate the partial-dual genus polynomials for some plane ribbon graphs. Additionally, we establish the asymptotic normality of the partial-dual genus distributions for specific plane ribbon graphs. Furthermore, we employ a plane ribbon graph to construct a minimal counterexample to the interpolation conjecture.