SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 828-844, March 2024. Abstract. We introduce the class of strong cocomparability graphs, as the class of reflexive graphs whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the submatrix with rows 01,10, which we call Slash. We provide an ordering characterization, a forbidden structure characterization, and a polynomial-time certifying recognition algorithm for the class. These results complete the picture in which in addition to, or instead of, the [math] matrix one forbids the [math] matrix (which has rows 11,10). It is well known that in these two cases one obtains the class of interval graphs and the class of strongly chordal graphs, respectively. By complementation, we obtain the class of strong comparability graphs, whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the two-by-two identity submatrix. Thus our results give characterizations and algorithms for this class of irreflexive graphs as well. In other words, our results may be interpreted as solving the following problem: given a symmetric 0,1-matrix with 0-diagonal, can the rows and columns of be simultaneously permuted to avoid the two-by-two identity submatrix?
{"title":"Strong Cocomparability Graphs and Slash-Free Orderings of Matrices","authors":"Pavol Hell, Jing Huang, Jephian C.-H. Lin","doi":"10.1137/22m153238x","DOIUrl":"https://doi.org/10.1137/22m153238x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 828-844, March 2024. <br/> Abstract. We introduce the class of strong cocomparability graphs, as the class of reflexive graphs whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the submatrix with rows 01,10, which we call Slash. We provide an ordering characterization, a forbidden structure characterization, and a polynomial-time certifying recognition algorithm for the class. These results complete the picture in which in addition to, or instead of, the [math] matrix one forbids the [math] matrix (which has rows 11,10). It is well known that in these two cases one obtains the class of interval graphs and the class of strongly chordal graphs, respectively. By complementation, we obtain the class of strong comparability graphs, whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the two-by-two identity submatrix. Thus our results give characterizations and algorithms for this class of irreflexive graphs as well. In other words, our results may be interpreted as solving the following problem: given a symmetric 0,1-matrix with 0-diagonal, can the rows and columns of be simultaneously permuted to avoid the two-by-two identity submatrix?","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 790-827, March 2024. Abstract. Let [math] be a set of points in [math], where each point [math] has an associated transmission range, denoted [math]. The range assignment [math] induces a directed communication graph [math] on [math], which contains an edge [math] iff [math]. In the broadcast range-assignment problem, the goal is to assign the ranges such that [math] contains an arborescence rooted at a designated root node and the cost [math] of the assignment is minimized. We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution—the number of ranges that are modified when a point is inserted into or deleted from [math]—and its approximation ratio. To this end we study [math]-stable algorithms, which are algorithms that modify the range of at most [math] points when they update the solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm [math] that, for any given fixed parameter [math], is [math]-stable and that maintains a solution with approximation ratio [math], where the stability parameter [math] only depends on [math] and not on the size of [math]. We study such trade-offs in three settings. (1) For the problem in [math], we present a SAS with [math]. Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have [math]. We also present 1-, 2-, and 3-stable algorithms with constant approximation ratio. (2) For the problem in [math] (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in [math], we prove that an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in [math]. (3) For the problem in [math], we also prove that no SAS exists, and we present a [math]-stable [math]-approximation algorithm. Most results generalize to the setting where, for any given constant [math], the range-assignment cost is [math].
{"title":"Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem","authors":"Mark de Berg, Arpan Sadhukhan, Frits Spieksma","doi":"10.1137/23m1545975","DOIUrl":"https://doi.org/10.1137/23m1545975","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 790-827, March 2024. <br/> Abstract. Let [math] be a set of points in [math], where each point [math] has an associated transmission range, denoted [math]. The range assignment [math] induces a directed communication graph [math] on [math], which contains an edge [math] iff [math]. In the broadcast range-assignment problem, the goal is to assign the ranges such that [math] contains an arborescence rooted at a designated root node and the cost [math] of the assignment is minimized. We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution—the number of ranges that are modified when a point is inserted into or deleted from [math]—and its approximation ratio. To this end we study [math]-stable algorithms, which are algorithms that modify the range of at most [math] points when they update the solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm [math] that, for any given fixed parameter [math], is [math]-stable and that maintains a solution with approximation ratio [math], where the stability parameter [math] only depends on [math] and not on the size of [math]. We study such trade-offs in three settings. (1) For the problem in [math], we present a SAS with [math]. Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have [math]. We also present 1-, 2-, and 3-stable algorithms with constant approximation ratio. (2) For the problem in [math] (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in [math], we prove that an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in [math]. (3) For the problem in [math], we also prove that no SAS exists, and we present a [math]-stable [math]-approximation algorithm. Most results generalize to the setting where, for any given constant [math], the range-assignment cost is [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"52 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 726-742, March 2024. Abstract. We construct explicit combinatorial Bernoulli factories for the following class of flow-based polytopes: integral 0/1-polytopes defined by a set of network flow constraints. This generalizes the results of Niazadeh et al. (who constructed an explicit factory for the specific case of bipartite perfect matchings) and provides novel exact sampling procedures for sampling paths, circulations, and [math]-flows. In the process, we uncover new connections to algebraic combinatorics.
{"title":"Bernoulli Factories for Flow-Based Polytopes","authors":"Rad Niazadeh, Renato Paes Leme, Jon Schneider","doi":"10.1137/23m1558343","DOIUrl":"https://doi.org/10.1137/23m1558343","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 726-742, March 2024. <br/> Abstract. We construct explicit combinatorial Bernoulli factories for the following class of flow-based polytopes: integral 0/1-polytopes defined by a set of network flow constraints. This generalizes the results of Niazadeh et al. (who constructed an explicit factory for the specific case of bipartite perfect matchings) and provides novel exact sampling procedures for sampling paths, circulations, and [math]-flows. In the process, we uncover new connections to algebraic combinatorics.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yann Disser, Max Klimm, Annette Lutz, David Weckbecker
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 764-789, March 2024. Abstract. We consider the problem of maximizing a fractionally subadditive function under an increasing knapsack constraint. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most [math], under the assumption that the values of singleton sets are in the range [math], and we give a lower bound of [math] on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a lower bound of [math] and an upper bound of [math] for the incremental maximization of classical flows with capacities in [math] which is tight for the unit capacity case.
{"title":"Fractionally Subadditive Maximization under an Incremental Knapsack Constraint with Applications to Incremental Flows","authors":"Yann Disser, Max Klimm, Annette Lutz, David Weckbecker","doi":"10.1137/23m1569265","DOIUrl":"https://doi.org/10.1137/23m1569265","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 764-789, March 2024. <br/> Abstract. We consider the problem of maximizing a fractionally subadditive function under an increasing knapsack constraint. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most [math], under the assumption that the values of singleton sets are in the range [math], and we give a lower bound of [math] on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a lower bound of [math] and an upper bound of [math] for the incremental maximization of classical flows with capacities in [math] which is tight for the unit capacity case.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
James Cruickshank, Fatemeh Mohammadi, Harshit J. Motwani, Anthony Nixon, Shin-ichi Tanigawa
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 743-763, March 2024. Abstract. We consider the global rigidity problem for bar-joint frameworks where each vertex is constrained to lie on a particular line in [math]. In our setting, we allow multiple vertices to be constrained to the same line. We give a combinatorial characterization of generic rigidity in this setting for arbitrary line sets. Further, under a mild assumption on the given set of lines, we give a complete combinatorial characterization of graphs that are generically globally rigid. This gives a [math]-dimensional extension of the well-known combinatorial characterization of two-dimensional global rigidity. In particular, our results imply that global rigidity is a generic property in this setting.
{"title":"Global Rigidity of Line Constrained Frameworks","authors":"James Cruickshank, Fatemeh Mohammadi, Harshit J. Motwani, Anthony Nixon, Shin-ichi Tanigawa","doi":"10.1137/22m151707x","DOIUrl":"https://doi.org/10.1137/22m151707x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 743-763, March 2024. <br/> Abstract. We consider the global rigidity problem for bar-joint frameworks where each vertex is constrained to lie on a particular line in [math]. In our setting, we allow multiple vertices to be constrained to the same line. We give a combinatorial characterization of generic rigidity in this setting for arbitrary line sets. Further, under a mild assumption on the given set of lines, we give a complete combinatorial characterization of graphs that are generically globally rigid. This gives a [math]-dimensional extension of the well-known combinatorial characterization of two-dimensional global rigidity. In particular, our results imply that global rigidity is a generic property in this setting.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"88 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 702-725, March 2024. Abstract. In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements [math] and [math] are placed in order [math] with probability [math]. Our goal is to identify the class of parameter sets [math] for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill [Background on the Gap Problem (2003) and An Interesting Spectral Gap Problem (2003)] that all monotone, positively biased distributions are rapidly mixing. We resolve Fill’s conjecture in the affirmative for distributions arising from [math]-class particle processes, where the elements are divided into [math] classes and the probability of exchanging neighboring elements depends on the particular classes the elements are in. We further require that [math] is a constant and that all probabilities between elements in different classes are bounded away from [math]. These particle processes arise in the context of self-organizing lists, and our result also applies beyond permutations to the setting where all particles in a class are indistinguishable. Our work generalizes recent work by Haddadan and Winkler [Mixing of permutations by biased transposition (2017)] studying 3-class particle processes. Additionally, we show that a broader class of distributions based on trees is also rapidly mixing, which generalizes a class analyzed by Bhakta et al. [Mixing times of Markov chains for self-organizing lists and biased permutations (2013)]. Our proof involves analyzing a generalized biased exclusion process, which is a nearest-neighbor transposition chain applied to a 2-particle system. Biased exclusion processes are of independent interest, with applications in self-assembly. We generalize the results of Greenberg et al. [Sampling biased lattice configurations using exponential metrics (2009)] and Benjamini et al. [Mixing times of the biased card shuffling and the asymmetric exclusion process (2005)] on biased exclusion processes to allow the probability of swapping neighboring elements to depend on the entire system, as long as the minimum bias is bounded away from 1.
SIAM 离散数学杂志》，第 38 卷第 1 期，第 702-725 页，2024 年 3 月。 摘要在本文中，我们研究了最近邻换位马尔可夫链在排列集合上的偏置版本，其中相邻元素[math]和[math]以概率[math]按顺序[math]排列。我们的目标是找出该马尔可夫链快速混合的参数集 [math] 类别。具体来说，我们考虑了吉姆-菲尔（Jim Fill）的公开猜想[《差距问题背景》（2003）和《有趣的谱差距问题》（2003）]，即所有单调、正偏分布都是快速混合的。对于[math]类粒子过程产生的分布，我们以肯定的态度解决了菲尔的猜想，在[math]类粒子过程中，元素被划分为[math]类，相邻元素交换的概率取决于元素所处的特定类。我们进一步要求[math]是一个常数，不同类别元素之间的所有概率都远离[math]。这些粒子过程是在自组织列表的背景下产生的，我们的结果也超越了排列，适用于一类中所有粒子都无法区分的情况。我们的工作概括了 Haddadan 和 Winkler [Mixing of permutations by biased transposition (2017)]最近研究 3 类粒子过程的工作。此外，我们还证明了基于树的一类更广泛的分布也是快速混合的，这也概括了 Bhakta 等人【自组织列表和偏置排列的马尔可夫链的混合时间（2013 年）】所分析的一类分布。我们的证明涉及对广义偏置排除过程的分析，该过程是应用于双粒子系统的近邻转置链。偏置排除过程具有独立的意义，可应用于自组装。我们推广了格林伯格等人（Greenberg et al. [Sampling biased lattice configurations using exponential metrics (2009)]）和本杰明尼等人（Benjamini et al. [Mixing times of the biased card shuffling and the asymmetric exclusion process (2005)]）关于偏置排除过程的研究成果，只要最小偏置离 1 有界，就允许交换相邻元素的概率取决于整个系统。
{"title":"Rapid Mixing of [math]-Class Biased Permutations","authors":"Sarah Miracle, Amanda Pascoe Streib","doi":"10.1137/22m148063x","DOIUrl":"https://doi.org/10.1137/22m148063x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 702-725, March 2024. <br/> Abstract. In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements [math] and [math] are placed in order [math] with probability [math]. Our goal is to identify the class of parameter sets [math] for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill [Background on the Gap Problem (2003) and An Interesting Spectral Gap Problem (2003)] that all monotone, positively biased distributions are rapidly mixing. We resolve Fill’s conjecture in the affirmative for distributions arising from [math]-class particle processes, where the elements are divided into [math] classes and the probability of exchanging neighboring elements depends on the particular classes the elements are in. We further require that [math] is a constant and that all probabilities between elements in different classes are bounded away from [math]. These particle processes arise in the context of self-organizing lists, and our result also applies beyond permutations to the setting where all particles in a class are indistinguishable. Our work generalizes recent work by Haddadan and Winkler [Mixing of permutations by biased transposition (2017)] studying 3-class particle processes. Additionally, we show that a broader class of distributions based on trees is also rapidly mixing, which generalizes a class analyzed by Bhakta et al. [Mixing times of Markov chains for self-organizing lists and biased permutations (2013)]. Our proof involves analyzing a generalized biased exclusion process, which is a nearest-neighbor transposition chain applied to a 2-particle system. Biased exclusion processes are of independent interest, with applications in self-assembly. We generalize the results of Greenberg et al. [Sampling biased lattice configurations using exponential metrics (2009)] and Benjamini et al. [Mixing times of the biased card shuffling and the asymmetric exclusion process (2005)] on biased exclusion processes to allow the probability of swapping neighboring elements to depend on the entire system, as long as the minimum bias is bounded away from 1.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 668-701, March 2024. Abstract. Szemerédi’s regularity lemma guarantees that, for fixed [math], every graph [math] admits an [math]-regular and [math]-equitable partition [math], where [math]. These partitions are constructed by Kohayakawa, Rödl, and Thoma in time [math]. Analogous partitions of [math]-graphs [math] are constructed by Czygrinow and Rödl in time [math]. For [math], we construct these partitions (and others with slightly stronger regularity) in time [math]. We also discuss some applications.
{"title":"Some Cubic Time Regularity Algorithms for Triple Systems","authors":"Brendan Nagle, John Theado","doi":"10.1137/21m145046x","DOIUrl":"https://doi.org/10.1137/21m145046x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 668-701, March 2024. <br/> Abstract. Szemerédi’s regularity lemma guarantees that, for fixed [math], every graph [math] admits an [math]-regular and [math]-equitable partition [math], where [math]. These partitions are constructed by Kohayakawa, Rödl, and Thoma in time [math]. Analogous partitions of [math]-graphs [math] are constructed by Czygrinow and Rödl in time [math]. For [math], we construct these partitions (and others with slightly stronger regularity) in time [math]. We also discuss some applications.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 629-644, March 2024. Abstract. In 1959, Erdős and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here, we study a rainbow version of their theorem, in which one considers [math] graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any [math].
{"title":"Graphs without a Rainbow Path of Length 3","authors":"Sebastian Babiński, Andrzej Grzesik","doi":"10.1137/22m1535048","DOIUrl":"https://doi.org/10.1137/22m1535048","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 629-644, March 2024. <br/> Abstract. In 1959, Erdős and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here, we study a rainbow version of their theorem, in which one considers [math] graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"17 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139670059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 645-667, March 2024. Abstract. Fix [math], and let [math] be a graph, with vertex set partitioned into [math] subsets (“blocks”) of approximately equal size. An induced subgraph of [math] is “transversal” (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly [math] vertices). A “pure pair” in [math] is a pair [math] of disjoint subsets of [math] such that either all edges between [math] are present or none are; and in the present context we are interested in pure pairs [math] where each of [math] is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.
{"title":"Pure Pairs. IX. Transversal Trees","authors":"Alex Scott, Paul Seymour, Sophie T. Spirkl","doi":"10.1137/21m1456509","DOIUrl":"https://doi.org/10.1137/21m1456509","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 645-667, March 2024. <br/> Abstract. Fix [math], and let [math] be a graph, with vertex set partitioned into [math] subsets (“blocks”) of approximately equal size. An induced subgraph of [math] is “transversal” (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly [math] vertices). A “pure pair” in [math] is a pair [math] of disjoint subsets of [math] such that either all edges between [math] are present or none are; and in the present context we are interested in pure pairs [math] where each of [math] is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"28 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139665329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 609-628, March 2024. Abstract. Hindman proved in 1979 that no matter how natural numbers are colored in [math] colors, for a fixed positive integer [math], there is an infinite subset [math] of numbers and a color [math] such that for any finite nonempty subset [math] of [math], the color of the sum of elements from [math] is [math]. Later, Taylor extended this result to colorings with an unrestricted number of colors and five unavoidable color patterns on finite sums. This result is referred to as a canonization of Hindman’s theorem and parallels the canonical Ramsey theorem of Erdős and Rado. We extend Taylor’s result from sums, that are linear combinations with coefficients 1, to several linear combinations with coefficients 1 and [math]. These results in turn could be interpreted as canonical-type theorems for solutions to infinite systems.
{"title":"Canonical Theorems for Colored Integers with Respect to Some Linear Combinations","authors":"Maria Axenovich, Hanno Lefmann","doi":"10.1137/21m1454195","DOIUrl":"https://doi.org/10.1137/21m1454195","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 609-628, March 2024. <br/> Abstract. Hindman proved in 1979 that no matter how natural numbers are colored in [math] colors, for a fixed positive integer [math], there is an infinite subset [math] of numbers and a color [math] such that for any finite nonempty subset [math] of [math], the color of the sum of elements from [math] is [math]. Later, Taylor extended this result to colorings with an unrestricted number of colors and five unavoidable color patterns on finite sums. This result is referred to as a canonization of Hindman’s theorem and parallels the canonical Ramsey theorem of Erdős and Rado. We extend Taylor’s result from sums, that are linear combinations with coefficients 1, to several linear combinations with coefficients 1 and [math]. These results in turn could be interpreted as canonical-type theorems for solutions to infinite systems.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"08 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139670126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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