Cycles in 3-connected claw-free planar graphs and 4-connected planar graphs without 4-cycles

IF 0.9 3区 数学 Q2 MATHEMATICS
On-Hei Solomon Lo
{"title":"Cycles in 3-connected claw-free planar graphs and 4-connected planar graphs without 4-cycles","authors":"On-Hei Solomon Lo","doi":"10.1002/jgt.23152","DOIUrl":null,"url":null,"abstract":"<p>The cycle spectrum <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>CS</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${\\mathscr{CS}}(G)$</annotation>\n </semantics></math> of a graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is the set of the cycle lengths in <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>. Let <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> ${\\mathscr{G}}$</annotation>\n </semantics></math> be a graph class. For any integer <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> $k\\ge 3$</annotation>\n </semantics></math>, define <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>f</mi>\n \n <mi>G</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>k</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${f}_{{\\mathscr{G}}}(k)$</annotation>\n </semantics></math> to be the least integer <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>k</mi>\n \n <mo>′</mo>\n </msup>\n \n <mo>≥</mo>\n \n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> ${k}^{^{\\prime} }\\ge k$</annotation>\n </semantics></math> such that for any <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n \n <mo>∈</mo>\n \n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G\\in {\\mathscr{G}}$</annotation>\n </semantics></math> with circumference at least <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>CS</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∩</mo>\n \n <mrow>\n <mo>[</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>,</mo>\n \n <msup>\n <mi>k</mi>\n \n <mo>′</mo>\n </msup>\n </mrow>\n \n <mo>]</mo>\n </mrow>\n \n <mo>≠</mo>\n \n <mi>∅</mi>\n </mrow>\n </mrow>\n <annotation> $k,{\\mathscr{CS}}(G)\\cap [k,{k}^{^{\\prime} }]\\ne \\varnothing $</annotation>\n </semantics></math>. Denote by <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>P</mi>\n \n <mo>,</mo>\n \n <mi>P</mi>\n \n <mn>3</mn>\n \n <mo>,</mo>\n \n <mi>PC</mi>\n </mrow>\n </mrow>\n <annotation> ${\\mathscr{P}},{\\mathscr{P}}3,{\\mathscr{PC}}$</annotation>\n </semantics></math>, and <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>PC</mi>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> ${\\mathscr{PC}}3$</annotation>\n </semantics></math> the classes of 3-connected planar graphs, 3-connected cubic planar graphs, 3-connected claw-free planar graphs, and 3-connected claw-free cubic planar graphs, respectively. The values of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>f</mi>\n \n <mi>P</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>k</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${f}_{{\\mathscr{P}}}(k)$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>f</mi>\n \n <mrow>\n <mi>P</mi>\n \n <mn>3</mn>\n </mrow>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>k</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${f}_{{\\mathscr{P}}3}(k)$</annotation>\n </semantics></math> were known for all <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> $k\\ge 3$</annotation>\n </semantics></math>. In the first part of this article, we prove the claw-free version of these results by giving the values of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>f</mi>\n \n <mi>PC</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>k</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${f}_{{\\mathscr{PC}}}(k)$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>f</mi>\n \n <mrow>\n <mi>PC</mi>\n \n <mn>3</mn>\n </mrow>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>k</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${f}_{{\\mathscr{PC}}3}(k)$</annotation>\n </semantics></math> for all <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> $k\\ge 3$</annotation>\n </semantics></math>. In the second part we study the cycle spectra of 4-connected planar graphs without 4-cycles. Bondy conjectured that every 4-connected planar graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> has all cycle lengths from 3 to <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n </mrow>\n </mrow>\n <annotation> $| V(G)| $</annotation>\n </semantics></math> except possibly one even length. The truth of this conjecture would imply that every 4-connected planar graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> without 4-cycles has all cycle lengths other than 4. It was already known that <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mn>3</mn>\n \n <mo>,</mo>\n \n <mn>5</mn>\n \n <mo>,</mo>\n \n <mo>…</mo>\n \n <mo>,</mo>\n \n <mn>9</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n \n <mo>∪</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mrow>\n <mo>⌊</mo>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n \n <mo>∕</mo>\n \n <mn>2</mn>\n </mrow>\n \n <mo>⌋</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mo>…</mo>\n \n <mo>,</mo>\n \n <mrow>\n <mo>⌈</mo>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n \n <mo>∕</mo>\n \n <mn>2</mn>\n </mrow>\n \n <mo>⌉</mo>\n </mrow>\n \n <mo>+</mo>\n \n <mn>3</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n \n <mo>∪</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n \n <mo>−</mo>\n \n <mn>7</mn>\n \n <mo>,</mo>\n \n <mo>…</mo>\n \n <mo>,</mo>\n \n <mo>∣</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n \n <mo>⊆</mo>\n \n <mi>CS</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $\\{3,5,\\ldots ,9\\}\\cup \\{\\lfloor | V(G)| \\unicode{x02215}2\\rfloor ,\\ldots ,\\lceil | V(G)| \\unicode{x02215}2\\rceil +3\\}\\cup \\{| V(G)| -7,\\ldots ,| V(G)| \\}\\subseteq {\\mathscr{CS}}(G)$</annotation>\n </semantics></math>. We prove that <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> can be embedded in the plane such that for any <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>∈</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mrow>\n <mo>⌊</mo>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n \n <mo>∕</mo>\n \n <mn>2</mn>\n </mrow>\n \n <mo>⌋</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mo>…</mo>\n \n <mo>,</mo>\n \n <mo>∣</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $k\\in \\{\\lfloor | V(G)| \\unicode{x02215}2\\rfloor ,\\ldots ,| V(G)| \\},G$</annotation>\n </semantics></math> has a <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-cycle <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>C</mi>\n </mrow>\n </mrow>\n <annotation> $C$</annotation>\n </semantics></math> and all vertices not in <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>C</mi>\n </mrow>\n </mrow>\n <annotation> $C$</annotation>\n </semantics></math> lie in the exterior of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>C</mi>\n </mrow>\n </mrow>\n <annotation> $C$</annotation>\n </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"702-728"},"PeriodicalIF":0.9000,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23152","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

The cycle spectrum CS ( G ) ${\mathscr{CS}}(G)$ of a graph G $G$ is the set of the cycle lengths in G $G$ . Let G ${\mathscr{G}}$ be a graph class. For any integer k 3 $k\ge 3$ , define f G ( k ) ${f}_{{\mathscr{G}}}(k)$ to be the least integer k k ${k}^{^{\prime} }\ge k$ such that for any G G $G\in {\mathscr{G}}$ with circumference at least k , CS ( G ) [ k , k ] $k,{\mathscr{CS}}(G)\cap [k,{k}^{^{\prime} }]\ne \varnothing $ . Denote by P , P 3 , PC ${\mathscr{P}},{\mathscr{P}}3,{\mathscr{PC}}$ , and PC 3 ${\mathscr{PC}}3$ the classes of 3-connected planar graphs, 3-connected cubic planar graphs, 3-connected claw-free planar graphs, and 3-connected claw-free cubic planar graphs, respectively. The values of f P ( k ) ${f}_{{\mathscr{P}}}(k)$ and f P 3 ( k ) ${f}_{{\mathscr{P}}3}(k)$ were known for all k 3 $k\ge 3$ . In the first part of this article, we prove the claw-free version of these results by giving the values of f PC ( k ) ${f}_{{\mathscr{PC}}}(k)$ and f PC 3 ( k ) ${f}_{{\mathscr{PC}}3}(k)$ for all k 3 $k\ge 3$ . In the second part we study the cycle spectra of 4-connected planar graphs without 4-cycles. Bondy conjectured that every 4-connected planar graph G $G$ has all cycle lengths from 3 to V ( G ) $| V(G)| $ except possibly one even length. The truth of this conjecture would imply that every 4-connected planar graph G $G$ without 4-cycles has all cycle lengths other than 4. It was already known that { 3 , 5 , , 9 } { V ( G ) 2 , , V ( G ) 2 + 3 } { V ( G ) 7 , , V ( G ) } CS ( G ) $\{3,5,\ldots ,9\}\cup \{\lfloor | V(G)| \unicode{x02215}2\rfloor ,\ldots ,\lceil | V(G)| \unicode{x02215}2\rceil +3\}\cup \{| V(G)| -7,\ldots ,| V(G)| \}\subseteq {\mathscr{CS}}(G)$ . We prove that G $G$ can be embedded in the plane such that for any k { V ( G ) 2 , , V ( G ) } , G $k\in \{\lfloor | V(G)| \unicode{x02215}2\rfloor ,\ldots ,| V(G)| \},G$ has a k $k$ -cycle C $C$ and all vertices not in C $C$ lie in the exterior of C $C$ .

3连通无爪平面图和4连通无4周期平面图中的周期
图的循环谱是图中循环长度的集合。设为一个图类。对于任意整数 , 定义为最小整数,使得对于任意周长至少为 .用 、 和 分别表示 3 连接平面图形类、3 连接立方平面图形类、3 连接无爪平面图形类和 3 连接无爪立方平面图形类。所有......的和值都是已知的。在本文的第一部分,我们通过给出 和 的值,证明了这些结果的无爪版本。在第二部分中,我们将研究无 4 循环的 4 连接平面图的循环谱。邦迪猜想每个 4 连接平面图都有从 3 到的所有循环长度,只有一个偶数长度除外。这一猜想的真实性意味着每个不含 4 循环的 4 连接平面图都具有 4 以外的所有循环长度。我们证明了可以嵌入平面图中,使得对于任意有一个循环且所有不在循环中的顶点都位于 .
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来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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