Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano
{"title":"随机扭曲超立方体的直径","authors":"Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano","doi":"10.1016/j.ejc.2024.104078","DOIUrl":null,"url":null,"abstract":"<div><div>The <span><math><mi>n</mi></math></span>-dimensional random twisted hypercube <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is constructed recursively by taking two instances of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>log</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>/</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> with high probability and at least <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></math></span>. We improve their upper bound by showing that <span><math><mrow><mi>diam</mi><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mfrac><mrow><mi>n</mi></mrow><mrow><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></mfrac></mrow></math></span> with high probability.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The diameter of randomly twisted hypercubes\",\"authors\":\"Lucas Aragão , Maurício Collares , Gabriel Dahia , João Pedro Marciano\",\"doi\":\"10.1016/j.ejc.2024.104078\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>The <span><math><mi>n</mi></math></span>-dimensional random twisted hypercube <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is constructed recursively by taking two instances of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>log</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>/</mo><mo>log</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> with high probability and at least <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></math></span>. We improve their upper bound by showing that <span><math><mrow><mi>diam</mi><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mfrac><mrow><mi>n</mi></mrow><mrow><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi></mrow></mfrac></mrow></math></span> with high probability.</div></div>\",\"PeriodicalId\":50490,\"journal\":{\"name\":\"European Journal of Combinatorics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-10-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"European Journal of Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S019566982400163X\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S019566982400163X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
n 维随机扭曲超立方体 Gn 是由两个具有任意联合分布的 Gn-1 实例，并在它们的顶点集之间添加一个随机完美匹配来递归构造的。Benjamini、Dikstein、Gross 和 Zhukovskii 证明了其直径为 O(nloglogn/loglogn)，且概率很高，至少为 (n-1)/log2n。我们通过证明 diam(Gn)=(1+o(1))nlog2n 的高概率，改进了他们的上限。
The -dimensional random twisted hypercube is constructed recursively by taking two instances of , with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is with high probability and at least . We improve their upper bound by showing that with high probability.
期刊介绍：
The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.
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