Mohammadreza Bidgoli, Ali Mohammadian, Behruz Tayfeh-Rezaie, Maksim Zhukovskii
{"title":"弱饱和稳定性阈值","authors":"Mohammadreza Bidgoli, Ali Mohammadian, Behruz Tayfeh-Rezaie, Maksim Zhukovskii","doi":"10.1002/jgt.23079","DOIUrl":null,"url":null,"abstract":"<p>We study the weak <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n \n <mi>s</mi>\n </msub>\n </mrow>\n <annotation> ${K}_{s}$</annotation>\n </semantics></math>-saturation number of the Erdős–Rényi random graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>p</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${\\mathbb{G}}(n,p)$</annotation>\n </semantics></math>, denoted by <span></span><math>\n <semantics>\n <mrow>\n <mtext>wsat</mtext>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>G</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>p</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mi>s</mi>\n </msub>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{wsat}({\\mathbb{G}}(n,p),{K}_{s})$</annotation>\n </semantics></math>, where <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n \n <mi>s</mi>\n </msub>\n </mrow>\n <annotation> ${K}_{s}$</annotation>\n </semantics></math> is the complete graph on <span></span><math>\n <semantics>\n <mrow>\n <mi>s</mi>\n </mrow>\n <annotation> $s$</annotation>\n </semantics></math> vertices. In 2017, Korándi and Sudakov proved that the weak <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n \n <mi>s</mi>\n </msub>\n </mrow>\n <annotation> ${K}_{s}$</annotation>\n </semantics></math>-saturation number of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n \n <mi>n</mi>\n </msub>\n </mrow>\n <annotation> ${K}_{n}$</annotation>\n </semantics></math> is stable, in the sense that it remains the same after removing edges with constant probability. In this paper, we prove that there exists a threshold for this stability property and give upper and lower bounds on the threshold. This generalizes the result of Korándi and Sudakov. A general upper bound on <span></span><math>\n <semantics>\n <mrow>\n <mtext>wsat</mtext>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>G</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>p</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mi>s</mi>\n </msub>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{wsat}({\\mathbb{G}}(n,p),{K}_{s})$</annotation>\n </semantics></math> is also provided.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 3","pages":"474-495"},"PeriodicalIF":0.9000,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Threshold for stability of weak saturation\",\"authors\":\"Mohammadreza Bidgoli, Ali Mohammadian, Behruz Tayfeh-Rezaie, Maksim Zhukovskii\",\"doi\":\"10.1002/jgt.23079\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study the weak <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>K</mi>\\n \\n <mi>s</mi>\\n </msub>\\n </mrow>\\n <annotation> ${K}_{s}$</annotation>\\n </semantics></math>-saturation number of the Erdős–Rényi random graph <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>p</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> ${\\\\mathbb{G}}(n,p)$</annotation>\\n </semantics></math>, denoted by <span></span><math>\\n <semantics>\\n <mrow>\\n <mtext>wsat</mtext>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>G</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>p</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>K</mi>\\n \\n <mi>s</mi>\\n </msub>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{wsat}({\\\\mathbb{G}}(n,p),{K}_{s})$</annotation>\\n </semantics></math>, where <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>K</mi>\\n \\n <mi>s</mi>\\n </msub>\\n </mrow>\\n <annotation> ${K}_{s}$</annotation>\\n </semantics></math> is the complete graph on <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>s</mi>\\n </mrow>\\n <annotation> $s$</annotation>\\n </semantics></math> vertices. In 2017, Korándi and Sudakov proved that the weak <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>K</mi>\\n \\n <mi>s</mi>\\n </msub>\\n </mrow>\\n <annotation> ${K}_{s}$</annotation>\\n </semantics></math>-saturation number of <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>K</mi>\\n \\n <mi>n</mi>\\n </msub>\\n </mrow>\\n <annotation> ${K}_{n}$</annotation>\\n </semantics></math> is stable, in the sense that it remains the same after removing edges with constant probability. In this paper, we prove that there exists a threshold for this stability property and give upper and lower bounds on the threshold. This generalizes the result of Korándi and Sudakov. A general upper bound on <span></span><math>\\n <semantics>\\n <mrow>\\n <mtext>wsat</mtext>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>G</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>p</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>K</mi>\\n \\n <mi>s</mi>\\n </msub>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{wsat}({\\\\mathbb{G}}(n,p),{K}_{s})$</annotation>\\n </semantics></math> is also provided.</p>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"106 3\",\"pages\":\"474-495\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-02-23\",\"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.23079\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23079","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
We study the weak -saturation number of the Erdős–Rényi random graph , denoted by , where is the complete graph on vertices. In 2017, Korándi and Sudakov proved that the weak -saturation number of is stable, in the sense that it remains the same after removing edges with constant probability. In this paper, we prove that there exists a threshold for this stability property and give upper and lower bounds on the threshold. This generalizes the result of Korándi and Sudakov. A general upper bound on is also provided.
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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.
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