横汉密尔顿环的通用性

IF 0.9 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2025-01-10 DOI:10.1112/blms.13223

Candida Bowtell, Patrick Morris, Yanitsa Pehova, Katherine Staden

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We show that <math>\n <semantics>\n <mi>G</mi>\n <annotation>$\\mathbf {G}$</annotation>\n </semantics></math> contains every Hamilton cycle pattern. That is, for every map <math>\n <semantics>\n <mrow>\n <mi>χ</mi>\n <mo>:</mo>\n <mo>[</mo>\n <mi>n</mi>\n <mo>]</mo>\n <mo>→</mo>\n <mo>[</mo>\n <mi>m</mi>\n <mo>]</mo>\n </mrow>\n <annotation>$\\chi: [n] \\rightarrow [m]$</annotation>\n </semantics></math> there is a Hamilton cycle whose <math>\n <semantics>\n <mi>i</mi>\n <annotation>$i$</annotation>\n </semantics></math>th edge lies in <math>\n <semantics>\n <msub>\n <mi>G</mi>\n <mrow>\n <mi>χ</mi>\n <mo>(</mo>\n <mi>i</mi>\n <mo>)</mo>\n </mrow>\n </msub>\n <annotation>$G_{\\chi (i)}$</annotation>\n </semantics></math>.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"711-729"},"PeriodicalIF":0.9000,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13223","citationCount":"0","resultStr":"{\"title\":\"Universality for transversal Hamilton cycles\",\"authors\":\"Candida Bowtell, Patrick Morris, Yanitsa Pehova, Katherine Staden\",\"doi\":\"10.1112/blms.13223\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n <mo>=</mo>\\n <mo>{</mo>\\n <msub>\\n <mi>G</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <mtext>…</mtext>\\n <mo>,</mo>\\n <msub>\\n <mi>G</mi>\\n <mi>m</mi>\\n </msub>\\n <mo>}</mo>\\n </mrow>\\n <annotation>$\\\\mathbf {G}=\\\\lbrace G_1, \\\\ldots, G_m\\\\rbrace$</annotation>\\n </semantics></math> be a graph collection on a common vertex set <math>\\n <semantics>\\n <mi>V</mi>\\n <annotation>$V$</annotation>\\n </semantics></math> of size <math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math> such that <math>\\n <semantics>\\n <mrow>\\n <mi>δ</mi>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>G</mi>\\n <mi>i</mi>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <mo>⩾</mo>\\n <mrow>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>+</mo>\\n <mi>o</mi>\\n <mrow>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <mi>n</mi>\\n <mo>/</mo>\\n <mn>2</mn>\\n </mrow>\\n <annotation>$\\\\delta (G_i) \\\\geqslant (1+o(1))n/2$</annotation>\\n </semantics></math> for every <math>\\n <semantics>\\n <mrow>\\n <mi>i</mi>\\n <mo>∈</mo>\\n <mo>[</mo>\\n <mi>m</mi>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$i \\\\in [m]$</annotation>\\n </semantics></math>. 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引用次数: 0

摘要

设{G = g1，…，gm }$\mathbf {G}=\lbrace G_1, \ldots, G_m\rbrace$是大小为n $n$的公共顶点集V $V$上的图集合，使得δ（g1）大于或等于（1 + 0 (1)）N / 2 $\delta (G_i) \geqslant (1+o(1))n/2$对于每个I∈[m] $i \in [m]$。我们证明了G $\mathbf {G}$包含了所有的Hamilton循环模式。也就是说，对于每个地图χ：[n]→[m] $\chi: [n] \rightarrow [m]$存在一个Hamilton循环，其i $i$边位于G χ (1) $G_{\chi (i)}$。

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Universality for transversal Hamilton cycles

Let $G = {G_{1}, …, G_{m}}$ be a graph collection on a common vertex set $V$ of size $n$ such that $δ (G_{i}) ⩾ (1 + o (1)) n / 2$ for every $i \in [m]$ . We show that $G$ contains every Hamilton cycle pattern. That is, for every map $χ : [n] \to [m]$ there is a Hamilton cycle whose $i$ th edge lies in $G_{χ (i)}$ .