{"title":"Separating path systems of almost linear size","authors":"Shoham Letzter","doi":"10.1090/tran/9187","DOIUrl":null,"url":null,"abstract":"<p>A <italic>separating path system</italic> for a graph <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a collection <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"script\">P</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathcal {P}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of paths in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that for every two edges <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"e\"> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding=\"application/x-tex\">e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=\"application/x-tex\">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there is a path in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"script\">P</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathcal {P}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that contains <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"e\"> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding=\"application/x-tex\">e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> but not <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=\"application/x-tex\">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that every <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vertex graph has a separating path system of size <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis n log Superscript asterisk Baseline n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:msup> <mml:mi>log</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo></mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(n \\log ^* n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This improves upon the previous best upper bound of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis n log n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mi>log</mml:mi> <mml:mo></mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(n \\log n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and makes progress towards a conjecture of Falgas-Ravry–Kittipassorn–Korándi–Letzter–Narayanan and Balogh–Csaba–Martin–Pluhár, according to which an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bound should hold.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9187","RegionNum":2,"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
Abstract
A separating path system for a graph GG is a collection P\mathcal {P} of paths in GG such that for every two edges ee and ff, there is a path in P\mathcal {P} that contains ee but not ff. We show that every nn-vertex graph has a separating path system of size O(nlog∗n)O(n \log ^* n). This improves upon the previous best upper bound of O(nlogn)O(n \log n), and makes progress towards a conjecture of Falgas-Ravry–Kittipassorn–Korándi–Letzter–Narayanan and Balogh–Csaba–Martin–Pluhár, according to which an O(n)O(n) bound should hold.
图 G 的分离路径系统是图 G 中路径 P \mathcal {P} 的集合，对于每两条边 e e 和 f f，P \mathcal {P} 中都有一条路径包含 e e 而不包含 f f。我们证明，每个 n 个顶点图都有一个大小为 O ( n log ∗ n ) O(n \log ^* n) 的分离路径系统。这改进了之前的最佳上界 O ( n log n ) O(n \log n) ，并在实现 Falgas-Ravry-Kittipassorn-Korándi-Letzter-Narayanan 和 Balogh-Csaba-Martin-Pluhár 的猜想方面取得了进展，根据该猜想，O ( n ) O(n) 界应该成立。
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