It was shown by Kutnar, Marušič and Zhang in 2012 that every connected vertex-transitive graph of order 10p, where p is a prime and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2924_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p\ne 7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≠</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>, contains a Hamilton path, except for graphs X arising from the action of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2924_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{PSL}(2, s^m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>PSL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <msup> <mi>s</mi> <mi>m</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on cosets of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2924_Article_IEq3.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}_s^m\rtimes \mathbb {Z}_{\frac{s^m-1}{10}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="double-struck">Z</mi> <mi>s</mi> <mi>m</mi> </msubsup> <mo>⋊</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mfrac> <mrow> <msup> <mi>s</mi> <mi>m</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> <mn>10</mn> </mfrac> </msub> </mrow> </math></EquationSource> </InlineEquation>, where s is a prime. In this paper, Hamilton cycles of these exceptions X will be found.

On Hamilton Paths in Vertex-Transitive Graphs of Order 10p

期刊论文
发表日期：

2025年4月27日

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Shaofei Du,
Wenjuan Luo,
Hao Yu

摘要

It was shown by Kutnar, Marušič and Zhang in 2012 that every connected vertex-transitive graph of order 10p, where p is a prime and $p\ne 7$ $p \neq 7$ , contains a Hamilton path, except for graphs X arising from the action of $\textrm{PSL}(2, s^m)$ $PSL (2, s^{m})$ on cosets of $\mathbb {Z}_s^m\rtimes \mathbb {Z}_{\frac{s^m-1}{10}}$ $Z_{s}^{m} ⋊ Z_{\frac{s^{m} - 1}{10}}$ , where s is a prime. In this paper, Hamilton cycles of these exceptions X will be found.