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A Precise Condition for Independent Transversals in Bipartite Covers 二方覆盖中独立横截面的精确条件
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-05-06 DOI: 10.1137/23m1600384
Stijn Cambie, Penny Haxell, Ross J. Kang, Ronen Wdowinski
{"title":"A Precise Condition for Independent Transversals in Bipartite Covers","authors":"Stijn Cambie, Penny Haxell, Ross J. Kang, Ronen Wdowinski","doi":"10.1137/23m1600384","DOIUrl":"https://doi.org/10.1137/23m1600384","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1451-1461, June 2024. <br/> Abstract. Given a bipartite graph [math] in which any vertex in [math] (resp., [math]) has degree at most [math] (resp., [math]), suppose there is a partition of [math] that is a refinement of the bipartition [math] such that the parts in [math] (resp., [math]) have size at least [math] (resp., [math]). We prove that the condition [math] is sufficient for the existence of an independent set of vertices of [math] that is simultaneously transversal to the partition and show, moreover, that this condition is sharp. This result is a bipartite refinement of two well-known results on independent transversals, one due to the second author and the other due to Szabó and Tardos.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883090","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}
引用次数: 0
A Menger-Type Theorem for Two Induced Paths 两条诱导路径的门格尔定理
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-05-03 DOI: 10.1137/23m1573082
Sandra Albrechtsen, Tony Huynh, Raphael W. Jacobs, Paul Knappe, Paul Wollan
{"title":"A Menger-Type Theorem for Two Induced Paths","authors":"Sandra Albrechtsen, Tony Huynh, Raphael W. Jacobs, Paul Knappe, Paul Wollan","doi":"10.1137/23m1573082","DOIUrl":"https://doi.org/10.1137/23m1573082","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1438-1450, June 2024. <br/>Abstract. We give an approximate Menger-type theorem for the case when a graph [math] contains two [math] paths [math] and [math] such that [math] is an induced subgraph of [math]. More generally, we prove that there exists a function [math], such that for every graph [math] and [math], either there exist two [math] paths [math] and [math] such that the distance between [math] and [math] is at least [math], or there exists [math] such that the ball of radius [math] centered at [math] intersects every [math] path.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"95 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833159","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}
引用次数: 0
Hypergraph Horn Functions 超图角函数
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-05-02 DOI: 10.1137/23m1569162
Kristóf Bérczi, Endre Boros, Kazuhisa Makino
{"title":"Hypergraph Horn Functions","authors":"Kristóf Bérczi, Endre Boros, Kazuhisa Makino","doi":"10.1137/23m1569162","DOIUrl":"https://doi.org/10.1137/23m1569162","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1417-1437, June 2024. <br/> Abstract. Horn functions form a subclass of Boolean functions possessing interesting structural and computational properties. These functions play a fundamental role in algebra, artificial intelligence, combinatorics, computer science, database theory, and logic. In the present paper, we introduce the subclass of hypergraph Horn functions that generalizes matroids and equivalence relations. We provide multiple characterizations of hypergraph Horn functions in terms of implicate-duality and the closure operator, which are, respectively, regarded as generalizations of matroid duality and the Mac Lane–Steinitz exchange property of matroid closure. We also study algorithmic issues on hypergraph Horn functions and show that the recognition problem (i.e., deciding if a given definite Horn CNF represents a hypergraph Horn function) and key realization (i.e., deciding if a given hypergraph is realized as a key set by a hypergraph Horn function) can be done in polynomial time, while implicate sets can be generated with polynomial delay.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"55 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833156","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}
引用次数: 0
Sárközy’s Theorem in Various Finite Field Settings 各种有限域设置中的萨尔科齐定理
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-04-29 DOI: 10.1137/23m1563256
Anqi Li, Lisa Sauermann
{"title":"Sárközy’s Theorem in Various Finite Field Settings","authors":"Anqi Li, Lisa Sauermann","doi":"10.1137/23m1563256","DOIUrl":"https://doi.org/10.1137/23m1563256","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1409-1416, June 2024. <br/>Abstract. In this paper, we strengthen a result by Green about an analogue of Sárközy’s theorem in the setting of polynomial rings [math]. In the integer setting, for a given polynomial [math] with constant term zero, (a generalization of) Sárközy’s theorem gives an upper bound on the maximum size of a subset [math] that does not contain distinct [math] satisfying [math] for some [math]. Green proved an analogous result with much stronger bounds in the setting of subsets [math] of the polynomial ring [math], but this result required the additional condition that the number of roots of the polynomial [math] be coprime to [math]. We generalize Green’s result, removing this condition. As an application, we also obtain a version of Sárközy’s theorem with similar strong bounds for subsets [math] for [math] for a fixed prime [math] and large [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833114","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}
引用次数: 0
Random Necklaces Require Fewer Cuts 随机项链所需的剪裁更少
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-04-26 DOI: 10.1137/22m1506699
Noga Alon, Dor Elboim, János Pach, Gábor Tardos
{"title":"Random Necklaces Require Fewer Cuts","authors":"Noga Alon, Dor Elboim, János Pach, Gábor Tardos","doi":"10.1137/22m1506699","DOIUrl":"https://doi.org/10.1137/22m1506699","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1381-1408, June 2024. <br/>Abstract. It is known that any open necklace with beads of [math] types, in which the number of beads of each type is divisible by [math], can be partitioned by at most [math] cuts into intervals that can be distributed into [math] collections, each containing the same number of beads of each type. This is tight for all values of [math] and [math]. Here, we consider the case of random necklaces, where the number of beads of each type is [math]. Then the minimum number of cuts required for a “fair” partition with the above property is a random variable [math]. We prove that for fixed [math] and large [math], this random variable is at least [math] with high probability. For [math], fixed [math], and large [math], we determine the asymptotic behavior of the probability that [math] for all values of [math]. We show that this probability is polynomially small when [math], is bounded away from zero when [math], and decays like [math] when [math]. We also show that for large [math], [math] is at most [math] with high probability and that for large [math] and large ratio [math], [math] is [math] with high probability.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"103 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799188","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}
引用次数: 0
On the Chromatic Number of Random Regular Hypergraphs 论随机正则超图的色度数
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-04-24 DOI: 10.1137/22m1544476
Patrick Bennett, Alan Frieze
{"title":"On the Chromatic Number of Random Regular Hypergraphs","authors":"Patrick Bennett, Alan Frieze","doi":"10.1137/22m1544476","DOIUrl":"https://doi.org/10.1137/22m1544476","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1369-1380, June 2024. <br/>Abstract. We estimate the likely values of the chromatic and independence numbers of the random [math]-uniform [math]-regular hypergraph on [math] vertices for fixed [math], large fixed [math], and [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799116","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}
引用次数: 0
Tropicalizing the Graph Profile of Some Almost-Stars 将一些近似恒星的图谱热带化
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-04-22 DOI: 10.1137/23m1594947
Maria Dascălu, Annie Raymond
{"title":"Tropicalizing the Graph Profile of Some Almost-Stars","authors":"Maria Dascălu, Annie Raymond","doi":"10.1137/23m1594947","DOIUrl":"https://doi.org/10.1137/23m1594947","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1351-1368, June 2024. <br/> Abstract. Many important problems in extremal combinatorics can be stated as certifying polynomial inequalities in graph homomorphism numbers, and in particular, many ask to certify pure binomial inequalities. For a fixed collection of graphs [math], the tropicalization of the graph profile of [math] essentially records all valid pure binomial inequalities involving graph homomorphism numbers for graphs in [math]. Building upon ideas and techniques described by Blekherman and Raymond in 2022, we compute the tropicalization of the graph profile for the graph containing a single vertex as well as stars where one edge is subdivided. This allows pure binomial inequalities in homomorphism numbers (or densities) for these graphs to be verified through an explicit linear program where the number of variables is equal to the number of edges in the biggest graph involved.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"36 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806478","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}
引用次数: 0
An Algorithmic Framework for Locally Constrained Homomorphisms 局部受限同态的算法框架
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-04-18 DOI: 10.1137/22m1513290
Laurent Bulteau, Konrad K. Dabrowski, Noleen Köhler, Sebastian Ordyniak, Daniël Paulusma
{"title":"An Algorithmic Framework for Locally Constrained Homomorphisms","authors":"Laurent Bulteau, Konrad K. Dabrowski, Noleen Köhler, Sebastian Ordyniak, Daniël Paulusma","doi":"10.1137/22m1513290","DOIUrl":"https://doi.org/10.1137/22m1513290","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1315-1350, June 2024. <br/> Abstract. A homomorphism [math] from a guest graph [math] to a host graph [math] is locally bijective, injective, or surjective if for every [math], the restriction of [math] to the neighbourhood of [math] is bijective, injective, or surjective, respectively. We prove a number of new FPT (fixed-parameter tractable), W[1]-hard, and paraNP-complete results for the corresponding decision problems LBHom, LIHom, and LSHom by considering a hierarchy of parameters of the guest graph [math]. In this way we strengthen several existing results. For our FPT results, we develop a new algorithmic framework that involves a general ILP (integer linear program) model. We also use our framework to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140614777","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}
引用次数: 0
Discursive Voter Models on the Supercritical Scale-Free Network 超临界无标度网络上的议论性选民模型
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-04-16 DOI: 10.1137/22m1544373
John Fernley
{"title":"Discursive Voter Models on the Supercritical Scale-Free Network","authors":"John Fernley","doi":"10.1137/22m1544373","DOIUrl":"https://doi.org/10.1137/22m1544373","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1285-1314, June 2024. <br/> Abstract. The voter model is a classical interacting particle system, modeling how global consensus is formed by local imitation. We analyze the time to consensus for a particular family of voter models when the underlying structure is a scale-free inhomogeneous random graph in the high edge–density regime, where this graph features a giant component. In this regime, we verify that the polynomial orders of consensus agree with those of their mean-field approximation in [A. Moinet, A. Barrat, and R. Pastor-Satorras, Phys. Rev. E, 98 (2018), 022303]. This “discursive” family of models has a symmetrized interaction to better model discussions and is indexed by a temperature parameter that, for certain parameters of the power law tail of the network’s degree distribution, is seen to produce two distinct phases of consensus speed. Our proofs rely on the well-known duality to coalescing random walks and a novel bound on the mixing time of these walks using the known fast mixing of the Erdős–Rényi giant subgraph. Unlike in the subcritical case [J. Fernley and M. Ortgiese, Random Structures Algorithms, 62 (2023), pp. 376–429], which requires tail exponent of the limiting degree distribution [math] as well as low edge density, in the giant component case, we also address the “ultrasmall world” power law exponents [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"28 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140614800","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}
引用次数: 0
Rainbow Even Cycles 彩虹均匀周期
IF 0.8 3区 数学
SIAM Journal on Discrete Mathematics Pub Date : 2024-04-09 DOI: 10.1137/23m1564808
Zichao Dong, Zijian Xu
{"title":"Rainbow Even Cycles","authors":"Zichao Dong, Zijian Xu","doi":"10.1137/23m1564808","DOIUrl":"https://doi.org/10.1137/23m1564808","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1269-1284, June 2024. <br/> Abstract. We prove that every family of (not necessarily distinct) even cycles [math] on some fixed [math]-vertex set has a rainbow even cycle (that is, a set of edges from distinct [math]’s, forming an even cycle). This resolves an open problem of Aharoni, Briggs, Holzman and Jiang. Moreover, the result is best possible for every positive integer [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"124 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140562990","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}
引用次数: 0
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