A loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum -degree condition guarantees the existence of a loose Hamilton cycle in a -uniform hypergraph. For
超图中的松散汉密尔顿循环是指覆盖所有顶点的边的循环序列，其中只有每两条连续的边相交，而且恰好在一个顶点上相交。考虑到狄拉克定理，我们自然会问，在 k$$ k $$均匀超图中，保证松散汉密尔顿循环存在的最小 d$$ d $$度条件是什么。对于 k=3$$ k=3 $$ 和每个 d∈{1,2}$ din left{1,2right} $$，这样的必要条件和充分条件是已知的。我们证明，这些结果与它们的稀疏随机类比结果遵循 "转移原则"。证明结合了图环境中的几个想法，并依赖于吸收方法。特别是，我们采用了 Kwan 和 Ferber 的一种新方法，通过收缩过程在稀疏超图的子图中找到吸收体。在 d=2$$ d=2$$ 的情况下，我们的发现是渐近最优的。
{"title":"Transference for loose Hamilton cycles in random 3-uniform hypergraphs","authors":"Kalina Petrova, Miloš Trujić","doi":"10.1002/rsa.21216","DOIUrl":"https://doi.org/10.1002/rsa.21216","url":null,"abstract":"A loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum <span data-altimg=\"/cms/asset/96bf6800-4a53-4fcf-aa7b-a11ff802fdc1/rsa21216-math-0001.png\"></span><mjx-container ctxtmenu_counter=\"1746\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21216-math-0001.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"d\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21216:rsa21216-math-0001\" display=\"inline\" location=\"graphic/rsa21216-math-0001.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"d\" data-semantic-type=\"identifier\">d</mi></mrow>$$ d $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-degree condition guarantees the existence of a loose Hamilton cycle in a <span data-altimg=\"/cms/asset/e14262ab-ac20-4578-be3f-bbe30fc08b73/rsa21216-math-0002.png\"></span><mjx-container ctxtmenu_counter=\"1747\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21216-math-0002.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"k\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21216:rsa21216-math-0002\" display=\"inline\" location=\"graphic/rsa21216-math-0002.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"k\" data-semantic-type=\"identifier\">k</mi></mrow>$$ k $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-uniform hypergraph. For <span data-altimg=\"/cms/asset/03c2dae8-8d2b-4c90-a6ec-2587395b51ad/rsa21216-math-0003.png\"></span><mjx-container ctxtmenu_counter=\"1748\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21216-math-0003.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Annika Heckel, Marc Kaufmann, Noela Müller, Matija Pasch
In [Trans. Am. Math. Soc. 375 (2022), no. 1, 627–668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let and let
We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra (SICOMP 2018) for Boolean functions to the case of real-valued functions
Frederik Garbe, Jan Hladký, Gábor Kun, Kristýna Pekárková
The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order have a particularly simple structure. Namely, almost every fiber of the disintegration of the permuton (say, along the x-axis) consists only of atoms, at most many, and this bound is sharp. We use this to give a simple proof of the “permutation removal lemma.”
排列极限理论引出了称为 permutons 的极限对象，它们是单位平方上的某些博尔量。我们证明，避开阶数 k$$ k $$ 的给定置换的置换子具有特别简单的结构。也就是说，几乎每条分解 permuton 的纤维（比如说，沿着 x 轴）都只由原子组成，最多只有 (k-1)$$ left(k-1right) $$ 个，而且这个约束是尖锐的。我们利用这一点给出了 "包络去除稃证 "的简单证明。
{"title":"On pattern-avoiding permutons","authors":"Frederik Garbe, Jan Hladký, Gábor Kun, Kristýna Pekárková","doi":"10.1002/rsa.21208","DOIUrl":"https://doi.org/10.1002/rsa.21208","url":null,"abstract":"The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order <math altimg=\"urn:x-wiley:rsa:media:rsa21208:rsa21208-math-0001\" display=\"inline\" location=\"graphic/rsa21208-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>k</mi>\u0000</mrow>\u0000$$ k $$</annotation>\u0000</semantics></math> have a particularly simple structure. Namely, almost every fiber of the disintegration of the permuton (say, along the x-axis) consists only of atoms, at most <math altimg=\"urn:x-wiley:rsa:media:rsa21208:rsa21208-math-0002\" display=\"inline\" location=\"graphic/rsa21208-math-0002.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mo stretchy=\"false\">(</mo>\u0000<mi>k</mi>\u0000<mo form=\"prefix\">−</mo>\u0000<mn>1</mn>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ left(k-1right) $$</annotation>\u0000</semantics></math> many, and this bound is sharp. We use this to give a simple proof of the “permutation removal lemma.”","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139667155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a coupling between two seemingly very different constructions of the standard additive coalescent, which describes the evolution of masses merging pairwise at rates proportional to their sums. The first construction, due to Aldous and Pitman, involves the components obtained by logging the Brownian continuum random tree (CRT) by a Poissonian rain on its skeleton as time increases. The second one, due to Bertoin, involves the excursions above its running infimum of a linear-drifted standard Brownian excursion as its drift decreases. Our main tool is the use of an exploration algorithm of the so-called cut-tree of the Brownian CRT, which is a tree that encodes the genealogy of the fragmentation of the CRT.
{"title":"Coupling Bertoin's and Aldous–Pitman's representations of the additive coalescent","authors":"Igor Kortchemski, Paul Thévenin","doi":"10.1002/rsa.21206","DOIUrl":"https://doi.org/10.1002/rsa.21206","url":null,"abstract":"We construct a coupling between two seemingly very different constructions of the standard additive coalescent, which describes the evolution of masses merging pairwise at rates proportional to their sums. The first construction, due to Aldous and Pitman, involves the components obtained by logging the Brownian continuum random tree (CRT) by a Poissonian rain on its skeleton as time increases. The second one, due to Bertoin, involves the excursions above its running infimum of a linear-drifted standard Brownian excursion as its drift decreases. Our main tool is the use of an exploration algorithm of the so-called cut-tree of the Brownian CRT, which is a tree that encodes the genealogy of the fragmentation of the CRT.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139465102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nemanja Draganić, Abhishek Methuku, David Munhá Correia, Benny Sudakov
How many edges in an -vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any -vertex graph with
{"title":"Cycles with many chords","authors":"Nemanja Draganić, Abhishek Methuku, David Munhá Correia, Benny Sudakov","doi":"10.1002/rsa.21207","DOIUrl":"https://doi.org/10.1002/rsa.21207","url":null,"abstract":"How many edges in an <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21207-math-0001.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0001\" display=\"inline\" location=\"graphic/rsa21207-math-0001.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" data-semantic-type=\"identifier\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21207-math-0002.png\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21207:rsa21207-math-0002\" display=\"inline\" location=\"graphic/rsa21207-math-0002.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"n\" data-semantic-type=\"identifier\">n</mi></mrow>$$ n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-vertex graph with <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21207-math-0003.png\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,6\" data-semantic-content=\"7\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"2 n Superscript 3 divided by 2\" data-semantic-type=\"","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139464834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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