SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2509-2529, September 2024. Abstract. Given some binary matrix [math], suppose we are presented with the collection of its rows and columns in independent arbitrary orderings. From this information, can we recover the unique original orderings and matrix? We present an algorithm that identifies whether there is a unique ordering associated with a set of rows and columns, and outputs either the unique correct orderings for the rows and columns or the full collection of all valid orderings and valid matrices. We show that there is a constant [math] such that the algorithm terminates in [math] time with high probability and in expectation for random [math] binary matrices with i.i.d. entries [math] such that [math] and [math].
{"title":"An Algorithm to Recover Shredded Random Matrices","authors":"Caelan Atamanchuk, Luc Devroye, Massimo Vicenzo","doi":"10.1137/23m1615784","DOIUrl":"https://doi.org/10.1137/23m1615784","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2509-2529, September 2024. <br/> Abstract. Given some binary matrix [math], suppose we are presented with the collection of its rows and columns in independent arbitrary orderings. From this information, can we recover the unique original orderings and matrix? We present an algorithm that identifies whether there is a unique ordering associated with a set of rows and columns, and outputs either the unique correct orderings for the rows and columns or the full collection of all valid orderings and valid matrices. We show that there is a constant [math] such that the algorithm terminates in [math] time with high probability and in expectation for random [math] binary matrices with i.i.d. entries [math] such that [math] and [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256699","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 3, Page 2468-2488, September 2024. Abstract. It is known that after an appropriate rescaling the maximum degree of the binomial random graph converges in distribution to a Gumbel random variable. The same holds true for the maximum number of common neighbors of a [math]-vertex set and for the maximum number of [math]-cliques sharing a single vertex. Can these results be generalized to the maximum number of extensions of a [math]-vertex set for any given way of extending a [math]-vertex set by an [math]-vertex set? In this paper, we generalize the abovementioned results to a class of “symmetric extensions” and show that the limit distribution is not necessarily from the Gumbel family.
{"title":"Maximum Number of Symmetric Extensions in Random Graphs","authors":"Stepan Vakhrushev, Maksim Zhukovskii","doi":"10.1137/23m1588706","DOIUrl":"https://doi.org/10.1137/23m1588706","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2468-2488, September 2024. <br/> Abstract. It is known that after an appropriate rescaling the maximum degree of the binomial random graph converges in distribution to a Gumbel random variable. The same holds true for the maximum number of common neighbors of a [math]-vertex set and for the maximum number of [math]-cliques sharing a single vertex. Can these results be generalized to the maximum number of extensions of a [math]-vertex set for any given way of extending a [math]-vertex set by an [math]-vertex set? In this paper, we generalize the abovementioned results to a class of “symmetric extensions” and show that the limit distribution is not necessarily from the Gumbel family.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256701","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 3, Page 2489-2508, September 2024. Abstract. We prove that for [math] with [math] and [math], there exist [math] and [math] such that for every [math], every [math]-vertex graph [math] with [math] and [math] contains an [math]th power of a Hamilton cycle. We also show that the minimum degree condition is asymptotically sharp for [math] and the [math] case was recently conjectured by Staden and Treglown.
{"title":"On Powers of Hamilton Cycles in Ramsey–Turán Theory","authors":"Ming Chen, Jie Han, Yantao Tang, Donglei Yang","doi":"10.1137/24m163709x","DOIUrl":"https://doi.org/10.1137/24m163709x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2489-2508, September 2024. <br/> Abstract. We prove that for [math] with [math] and [math], there exist [math] and [math] such that for every [math], every [math]-vertex graph [math] with [math] and [math] contains an [math]th power of a Hamilton cycle. We also show that the minimum degree condition is asymptotically sharp for [math] and the [math] case was recently conjectured by Staden and Treglown.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256700","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 3, Page 2447-2467, September 2024. Abstract. In 1996 Guiduli and Mohar proposed a conjecture that predicts the structure of connected graphs with minimum degree [math] and minimum algebraic connectivity. We settle this conjecture for the case [math]. As a result, we conclude that the minimum algebraic connectivity of connected graphs with [math] vertices and [math] is [math], where [math] is a function in [math] that tends to 0 as [math] goes to infinity. This enables us to provide a positive answer to the problem of whether graphs with [math] and nearly maximum diameter have asymptotically minimum algebraic connectivity.
{"title":"Graphs of Degree at Least [math] with Minimum Algebraic Connectivity","authors":"Maryam Abdi, Ebrahim Ghorbani","doi":"10.1137/23m1585659","DOIUrl":"https://doi.org/10.1137/23m1585659","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2447-2467, September 2024. <br/> Abstract. In 1996 Guiduli and Mohar proposed a conjecture that predicts the structure of connected graphs with minimum degree [math] and minimum algebraic connectivity. We settle this conjecture for the case [math]. As a result, we conclude that the minimum algebraic connectivity of connected graphs with [math] vertices and [math] is [math], where [math] is a function in [math] that tends to 0 as [math] goes to infinity. This enables us to provide a positive answer to the problem of whether graphs with [math] and nearly maximum diameter have asymptotically minimum algebraic connectivity.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256800","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 3, Page 2429-2446, September 2024. Abstract. The [math]-blow-up of a given graph is obtained by replacing each edge by a clique of order [math] where the new vertices of the cliques are distinct. Liu and Yuan determined the extremal graphs for the 3-blow-ups of a triangle and the [math]-blow-ups of any complete graph with order at most [math], respectively. We determine the Turán number for the [math]-blow-ups of a complete graph with order at least [math], completing the study of the extremal graphs for [math]-blow-ups of complete graphs.
{"title":"On the Turán Number of Edge Blow-Ups of Cliques","authors":"Jialei Song, Changhong Lu, Long-Tu Yuan","doi":"10.1137/23m1623240","DOIUrl":"https://doi.org/10.1137/23m1623240","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2429-2446, September 2024. <br/> Abstract. The [math]-blow-up of a given graph is obtained by replacing each edge by a clique of order [math] where the new vertices of the cliques are distinct. Liu and Yuan determined the extremal graphs for the 3-blow-ups of a triangle and the [math]-blow-ups of any complete graph with order at most [math], respectively. We determine the Turán number for the [math]-blow-ups of a complete graph with order at least [math], completing the study of the extremal graphs for [math]-blow-ups of complete graphs.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202427","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 3, Page 2416-2428, September 2024. Abstract. In 1984, Erdős conjectured that the number of pentagons in any triangle-free graph on [math] vertices is at most [math], which is sharp by the balanced blow-up of a pentagon. This was proved by Grzesik, and independently by Hatami et al. As an extension of this result for longer cycles, we prove that for each odd [math], the balanced blow-up of [math] (uniquely) maximizes the number of [math]-cycles among [math]-free graphs on [math] vertices, as long as [math] is sufficiently large. We also show that this is no longer true if [math] is not assumed to be sufficiently large. Our result strengthens results of Grzesik and Kielak who proved that for each odd [math], the balanced blow-up of [math] maximizes the number of [math]-cycles among graphs with a given number of vertices and no odd cycles of length less than [math]. We further show that if [math] and [math] are odd and [math] is sufficiently large compared to [math], then the balanced blow-up of [math] does not asymptotically maximize the number of [math]-cycles among [math]-free graphs on [math] vertices. This disproves a conjecture of Grzesik and Kielak.
{"title":"On the Generalized Turán Problem for Odd Cycles","authors":"Csongor Beke, Oliver Janzer","doi":"10.1137/24m1632632","DOIUrl":"https://doi.org/10.1137/24m1632632","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2416-2428, September 2024. <br/> Abstract. In 1984, Erdős conjectured that the number of pentagons in any triangle-free graph on [math] vertices is at most [math], which is sharp by the balanced blow-up of a pentagon. This was proved by Grzesik, and independently by Hatami et al. As an extension of this result for longer cycles, we prove that for each odd [math], the balanced blow-up of [math] (uniquely) maximizes the number of [math]-cycles among [math]-free graphs on [math] vertices, as long as [math] is sufficiently large. We also show that this is no longer true if [math] is not assumed to be sufficiently large. Our result strengthens results of Grzesik and Kielak who proved that for each odd [math], the balanced blow-up of [math] maximizes the number of [math]-cycles among graphs with a given number of vertices and no odd cycles of length less than [math]. We further show that if [math] and [math] are odd and [math] is sufficiently large compared to [math], then the balanced blow-up of [math] does not asymptotically maximize the number of [math]-cycles among [math]-free graphs on [math] vertices. This disproves a conjecture of Grzesik and Kielak.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202428","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}
Benjamin Merlin Bumpus, Bart M. P. Jansen, Jari J. H. de Kroon
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2392-2415, September 2024. Abstract. We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a nonempty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a [math]-essential vertex as one that is contained in all [math]-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of nonessential vertices in the solution.
{"title":"Search-Space Reduction via Essential Vertices","authors":"Benjamin Merlin Bumpus, Bart M. P. Jansen, Jari J. H. de Kroon","doi":"10.1137/23m1589347","DOIUrl":"https://doi.org/10.1137/23m1589347","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2392-2415, September 2024. <br/> Abstract. We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a nonempty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a [math]-essential vertex as one that is contained in all [math]-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of nonessential vertices in the solution.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202429","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}
Jiangdong Ai, Stefanie Gerke, Gregory Gutin, Anders Yeo, Yacong Zhou
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2370-2391, September 2024. Abstract. We obtain lower and upper bounds for the maximum weight of a directed cut in the classes of weighted digraphs and weighted acyclic digraphs as well as in some of their subclasses. We compare our results with those obtained for the maximum size of a directed cut in unweighted digraphs. In particular, we show that a lower bound obtained by Alon, Bollobás, Gyárfás, Lehel, and Scott [J. Graph Theory, 55 (2007), pp. 1–13] for unweighted acyclic digraphs can be extended to weighted digraphs with the maximum length of a cycle being bounded by a constant and the weight of every arc being at least one. We state a number of open problems.
{"title":"Bounds on Maximum Weight Directed Cut","authors":"Jiangdong Ai, Stefanie Gerke, Gregory Gutin, Anders Yeo, Yacong Zhou","doi":"10.1137/23m1567394","DOIUrl":"https://doi.org/10.1137/23m1567394","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2370-2391, September 2024. <br/> Abstract. We obtain lower and upper bounds for the maximum weight of a directed cut in the classes of weighted digraphs and weighted acyclic digraphs as well as in some of their subclasses. We compare our results with those obtained for the maximum size of a directed cut in unweighted digraphs. In particular, we show that a lower bound obtained by Alon, Bollobás, Gyárfás, Lehel, and Scott [J. Graph Theory, 55 (2007), pp. 1–13] for unweighted acyclic digraphs can be extended to weighted digraphs with the maximum length of a cycle being bounded by a constant and the weight of every arc being at least one. We state a number of open problems.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202430","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 3, Page 2361-2369, September 2024. Abstract. We extend a recent breakthrough result relating expectation thresholds and actual thresholds to include some rainbow versions.
{"title":"Rainbow Thresholds","authors":"Tolson Bell, Alan Frieze, Trent G. Marbach","doi":"10.1137/21m1425736","DOIUrl":"https://doi.org/10.1137/21m1425736","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2361-2369, September 2024. <br/> Abstract. We extend a recent breakthrough result relating expectation thresholds and actual thresholds to include some rainbow versions.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202432","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 3, Page 2312-2334, September 2024. Abstract. The semi-random graph process is a single player game in which the player is initially presented an empty graph on [math] vertices. In each round, a vertex [math] is presented to the player independently and uniformly at random. The player then adaptively selects a vertex [math] and adds the edge [math] to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. In this paper, we investigate the following three properties: containing a complete graph of order [math], having the chromatic number at least [math], and not having an independent set of size at least [math].
{"title":"Cliques, Chromatic Number, and Independent Sets in the Semi-random Process","authors":"David Gamarnik, Mihyun Kang, Paweł Prałat","doi":"10.1137/23m1561105","DOIUrl":"https://doi.org/10.1137/23m1561105","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2312-2334, September 2024. <br/> Abstract. The semi-random graph process is a single player game in which the player is initially presented an empty graph on [math] vertices. In each round, a vertex [math] is presented to the player independently and uniformly at random. The player then adaptively selects a vertex [math] and adds the edge [math] to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. In this paper, we investigate the following three properties: containing a complete graph of order [math], having the chromatic number at least [math], and not having an independent set of size at least [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202434","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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