We prove that for any generating set S of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2024_2336_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb Z}^{n}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>, the continuous edge chromatic number χ′c(G) of the Schreier graph of the Bernoulli shift action <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2024_2336_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="108" /> </InlineMediaObject> <EquationSource Format="TEX">$G=F(S, \ 2^{{\mathbb Z}^{n}})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>G</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <msup> <mn>2</mn> <mrow> <msup> <mrow> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is χ′(G) + 1 = ∣S∣ + 1. In particular, for the standard generating set, the continuous edge chromatic number of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2024_2336_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$F(2^{{\mathbb Z}^{n}})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>F</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow> <msup> <mrow> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is 2n + 1.

Continuous edge chromatic numbers of abelian group actions

期刊论文
发表日期：

2025年1月3日

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Su Gao,
Ruijun Wang,
Tianhao Wang

摘要

We prove that for any generating set S of ${\mathbb Z}^{n}$ $Z^{n}$ , the continuous edge chromatic number χ′_c(G) of the Schreier graph of the Bernoulli shift action $G=F(S, \ 2^{{\mathbb Z}^{n}})$ $G = F (S, 2^{Z^{n}})$ is χ′(G) + 1 = ∣S∣ + 1. In particular, for the standard generating set, the continuous edge chromatic number of $F(2^{{\mathbb Z}^{n}})$ $F (2^{Z^{n}})$ is 2n + 1.