In a recent paper, Hou et al. conjectured that there exist no regular maps of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2093_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$2^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and of type <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2093_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\{2^k,2^s\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msup> <mn>2</mn> <mi>k</mi> </msup> <mo>,</mo> <msup> <mn>2</mn> <mi>s</mi> </msup> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, where n, k, and s are positive integers satisfying <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2093_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">$2\le s<k<n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>s</mi> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2093_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$s+k>n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>+</mo> <mi>k</mi> <mo>></mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we give an affirmative answer to this conjecture.

Nonexistence of certain regular maps of 2-power order

期刊论文
发表日期：

2025年1月17日

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Yao Tian,
Xiaogang Li

摘要

In a recent paper, Hou et al. conjectured that there exist no regular maps of order $2^n$ $2^{n}$ and of type $\{2^k,2^s\}$ ${2^{k}, 2^{s}}$ , where n, k, and s are positive integers satisfying $2\le s<k<n-1$ $2 \leq s < k < n - 1$ and $s+k>n$ $s + k > n$ . In this paper, we give an affirmative answer to this conjecture.