Igor Araujo, Simón Piga, Andrew Treglown, Zimu Xiang
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1808-1839, June 2024. Abstract. Given graphs [math] and [math], a perfect [math]-tiling in [math] is a collection of vertex-disjoint copies of [math] in [math] that together cover all the vertices in [math]. The study of the minimum degree threshold forcing a perfect [math]-tiling in a graph [math] has a long history, culminating in the Kühn–Osthus theorem [D. Kühn and D. Osthus, Combinatorica, 29 (2009), pp. 65–107] which resolves this problem, up to an additive constant, for all graphs [math]. In this paper we initiate the study of the analogous question for edge-ordered graphs. In particular, we characterize for which edge-ordered graphs [math] this problem is well-defined. We also apply the absorbing method to asymptotically determine the minimum degree threshold for forcing a perfect [math]-tiling in an edge-ordered graph, where [math] is any fixed monotone path.
{"title":"Tiling Edge-Ordered Graphs with Monotone Paths and Other Structures","authors":"Igor Araujo, Simón Piga, Andrew Treglown, Zimu Xiang","doi":"10.1137/23m1572519","DOIUrl":"https://doi.org/10.1137/23m1572519","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1808-1839, June 2024. <br/> Abstract. Given graphs [math] and [math], a perfect [math]-tiling in [math] is a collection of vertex-disjoint copies of [math] in [math] that together cover all the vertices in [math]. The study of the minimum degree threshold forcing a perfect [math]-tiling in a graph [math] has a long history, culminating in the Kühn–Osthus theorem [D. Kühn and D. Osthus, Combinatorica, 29 (2009), pp. 65–107] which resolves this problem, up to an additive constant, for all graphs [math]. In this paper we initiate the study of the analogous question for edge-ordered graphs. In particular, we characterize for which edge-ordered graphs [math] this problem is well-defined. We also apply the absorbing method to asymptotically determine the minimum degree threshold for forcing a perfect [math]-tiling in an edge-ordered graph, where [math] is any fixed monotone path.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529751","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 2, Page 1784-1807, June 2024. Abstract. For a locally finite set, [math], the [math]th Brillouin zone of [math] is the region of points [math] for which [math] is the [math]th smallest among the Euclidean distances between [math] and the points in [math]. If [math] is a lattice, the [math]th Brillouin zones of the points in [math] are translates of each other, and together they tile space. Depending on the value of [math], they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in [math], and the convergence of the maximum volume of a chamber to zero for the integer lattice.
{"title":"Brillouin Zones of Integer Lattices and Their Perturbations","authors":"Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian, Mathijs Wintraecken","doi":"10.1137/22m1489071","DOIUrl":"https://doi.org/10.1137/22m1489071","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1784-1807, June 2024. <br/> Abstract. For a locally finite set, [math], the [math]th Brillouin zone of [math] is the region of points [math] for which [math] is the [math]th smallest among the Euclidean distances between [math] and the points in [math]. If [math] is a lattice, the [math]th Brillouin zones of the points in [math] are translates of each other, and together they tile space. Depending on the value of [math], they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in [math], and the convergence of the maximum volume of a chamber to zero for the integer lattice.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"13 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526707","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 2, Page 1676-1685, June 2024. Abstract. Tuza [Finite and Infinite Sets, Proc. Colloq. Math. Soc. János Bolyai 37, North Holland, 1981, p. 888] conjectured that [math] for all graphs [math], where [math] is the minimum size of an edge set whose removal makes [math] triangle-free and [math] is the maximum size of a collection of pairwise edge-disjoint triangles. Here, we generalize Tuza’s conjecture to simple binary matroids that do not contain the Fano plane as a restriction and prove that the geometric version of the conjecture holds for cographic matroids.
{"title":"Tuza’s Conjecture for Binary Geometries","authors":"Kazuhiro Nomoto, Jorn van der Pol","doi":"10.1137/22m1511229","DOIUrl":"https://doi.org/10.1137/22m1511229","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1676-1685, June 2024. <br/> Abstract. Tuza [Finite and Infinite Sets, Proc. Colloq. Math. Soc. János Bolyai 37, North Holland, 1981, p. 888] conjectured that [math] for all graphs [math], where [math] is the minimum size of an edge set whose removal makes [math] triangle-free and [math] is the maximum size of a collection of pairwise edge-disjoint triangles. Here, we generalize Tuza’s conjecture to simple binary matroids that do not contain the Fano plane as a restriction and prove that the geometric version of the conjecture holds for cographic matroids.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"71 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141173347","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}
Margaret Bayer, Mark Denker, Marija Jelić Milutinović, Rowan Rowlands, Sheila Sundaram, Lei Xue
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1630-1675, June 2024. Abstract. We define the [math]-cut complex of a graph [math] with vertex set [math] to be the simplicial complex whose facets are the complements of sets of size [math] in [math] inducing disconnected subgraphs of [math]. This generalizes the Alexander dual of a graph complex studied by Fröberg [Topics in Algebra, Part 2, PWN, Warsaw, 1990, pp. 57–70] and Eagon and Reiner [J. Pure Appl. Algebra, 130 (1998), pp. 265–275]. We describe the effect of various graph operations on the cut complex and study its shellability, homotopy type, and homology for various families of graphs, including trees, cycles, complete multipartite graphs, and the prism [math], using techniques from algebraic topology, discrete Morse theory, and equivariant poset topology.
{"title":"Topology of Cut Complexes of Graphs","authors":"Margaret Bayer, Mark Denker, Marija Jelić Milutinović, Rowan Rowlands, Sheila Sundaram, Lei Xue","doi":"10.1137/23m1569034","DOIUrl":"https://doi.org/10.1137/23m1569034","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1630-1675, June 2024. <br/>Abstract. We define the [math]-cut complex of a graph [math] with vertex set [math] to be the simplicial complex whose facets are the complements of sets of size [math] in [math] inducing disconnected subgraphs of [math]. This generalizes the Alexander dual of a graph complex studied by Fröberg [Topics in Algebra, Part 2, PWN, Warsaw, 1990, pp. 57–70] and Eagon and Reiner [J. Pure Appl. Algebra, 130 (1998), pp. 265–275]. We describe the effect of various graph operations on the cut complex and study its shellability, homotopy type, and homology for various families of graphs, including trees, cycles, complete multipartite graphs, and the prism [math], using techniques from algebraic topology, discrete Morse theory, and equivariant poset topology.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146393","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 2, Page 1586-1629, June 2024. Abstract. A Sidorenko bigraph is one whose density in a bigraphon [math] is minimized precisely when [math] is constant. Several techniques in the literature to prove the Sidorenko property consist of decomposing (typically in a tree decomposition) the bigraph into smaller building blocks with stronger properties. One prominent such technique is that of [math]-decompositions of Conlon and Lee, which uses weakly Hölder (or weakly norming) bigraphs as building blocks. In turn, to obtain weakly Hölder bigraphs, it is typical to use the chain of implications reflection bigraph [math] cut-percolating bigraph [math] weakly Hölder bigraph. In an earlier result by the author with Razborov, we provided a generalization of [math]-decompositions, called reflective tree decompositions, that uses much weaker building blocks, called induced-Sidorenko bigraphs, to also obtain Sidorenko bigraphs. In this paper, we show that “left-sided” versions of the concepts of reflection bigraph and cut-percolating bigraph yield a similar chain of implications: left-reflection bigraph [math] left-cut-percolating bigraph [math] induced-Sidorenko bigraph. We also show that under mild hypotheses the “left-sided” analogue of the weakly Hölder property (which is also obtained via a similar chain of implications) can be used to improve bounds on another result of Conlon and Lee that roughly says that bigraphs with enough vertices on the right side of each realized degree have the Sidorenko property.
{"title":"Left-Cut-Percolation and Induced-Sidorenko Bigraphs","authors":"Leonardo N. Coregliano","doi":"10.1137/22m1526794","DOIUrl":"https://doi.org/10.1137/22m1526794","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1586-1629, June 2024. <br/> Abstract. A Sidorenko bigraph is one whose density in a bigraphon [math] is minimized precisely when [math] is constant. Several techniques in the literature to prove the Sidorenko property consist of decomposing (typically in a tree decomposition) the bigraph into smaller building blocks with stronger properties. One prominent such technique is that of [math]-decompositions of Conlon and Lee, which uses weakly Hölder (or weakly norming) bigraphs as building blocks. In turn, to obtain weakly Hölder bigraphs, it is typical to use the chain of implications reflection bigraph [math] cut-percolating bigraph [math] weakly Hölder bigraph. In an earlier result by the author with Razborov, we provided a generalization of [math]-decompositions, called reflective tree decompositions, that uses much weaker building blocks, called induced-Sidorenko bigraphs, to also obtain Sidorenko bigraphs. In this paper, we show that “left-sided” versions of the concepts of reflection bigraph and cut-percolating bigraph yield a similar chain of implications: left-reflection bigraph [math] left-cut-percolating bigraph [math] induced-Sidorenko bigraph. We also show that under mild hypotheses the “left-sided” analogue of the weakly Hölder property (which is also obtained via a similar chain of implications) can be used to improve bounds on another result of Conlon and Lee that roughly says that bigraphs with enough vertices on the right side of each realized degree have the Sidorenko property.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153959","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}
Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr, Alexander Wolff
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1537-1565, June 2024. Abstract. An obstacle representation of a graph [math] consists of a set of pairwise disjoint simply connected closed regions and a one-to-one mapping of the vertices of [math] to points such that two vertices are adjacent in [math] if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each [math]-vertex graph is [math] [M. Balko, J. Cibulka, and P. Valtr, Discrete Comput. Geom., 59 (2018), pp. 143–164] and that there are [math]-vertex graphs whose obstacle number is [math] [V. Dujmović and P. Morin, Electron. J. Combin., 22 (2015), 3.1]. We improve this lower bound to [math] for simple polygons and to [math] for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of [math]-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some [math]-vertex graph is given as part of the input, then for some drawings [math] obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph [math] is fixed-parameter tractable in the vertex cover number of [math]. Second, we show that, given a graph [math] and a simple polygon [math], it is NP-hard to decide whether [math] admits an obstacle representation using [math] as the only obstacle.
SIAM 离散数学杂志》，第 38 卷第 2 期，第 1537-1565 页，2024 年 6 月。 摘要。一个图[math]的障碍表示由一组成对不相交的简单连接封闭区域和[math]顶点到点的一一映射组成，当且仅当连接两个对应点的线段不与任何障碍相交时，两个顶点在[math]中相邻。图形的障碍数是图形在平面上的障碍表示中，所有障碍都是简单多边形的最小障碍数。已知每个[math]-顶点图的障碍数为[math] [M. Balko, J. Cibibi, J. Cibibi, J. M.Balko, J. Cibulka, and P. Valtr, Discrete Comput.Geom., 59 (2018)，第 143-164 页]，并且存在障碍数为[math]的[math]-顶点图[V. Dujmović and P. Morin, Electron. J. Combin., 22 (2015)，3.1]。对于简单多边形，我们将这一下界改进为[math]；对于凸多边形，我们将其改进为[math]。为了得到这些更强的下界，我们改进了对障碍数有界的[math]顶点图数量的已知估计，解决了杜伊莫维奇和莫林的一个猜想。我们还证明了，如果把某个[数学]顶点图的绘制作为输入的一部分，那么对于某些绘制来说，需要[数学]障碍才能把它们变成图的障碍表示。我们的边界在一些情况下是渐近紧密的。我们用两个复杂度结果来补充这些组合界限。首先，我们证明了计算一个图[math]的障碍数在[math]的顶点覆盖数中是固定参数可控的。其次，我们证明，给定一个图 [math] 和一个简单多边形 [math]，用 [math] 作为唯一的障碍来决定 [math] 是否允许障碍表示是 NP 难的。
{"title":"Bounding and Computing Obstacle Numbers of Graphs","authors":"Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr, Alexander Wolff","doi":"10.1137/23m1585088","DOIUrl":"https://doi.org/10.1137/23m1585088","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1537-1565, June 2024. <br/> Abstract. An obstacle representation of a graph [math] consists of a set of pairwise disjoint simply connected closed regions and a one-to-one mapping of the vertices of [math] to points such that two vertices are adjacent in [math] if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each [math]-vertex graph is [math] [M. Balko, J. Cibulka, and P. Valtr, Discrete Comput. Geom., 59 (2018), pp. 143–164] and that there are [math]-vertex graphs whose obstacle number is [math] [V. Dujmović and P. Morin, Electron. J. Combin., 22 (2015), 3.1]. We improve this lower bound to [math] for simple polygons and to [math] for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of [math]-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some [math]-vertex graph is given as part of the input, then for some drawings [math] obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph [math] is fixed-parameter tractable in the vertex cover number of [math]. Second, we show that, given a graph [math] and a simple polygon [math], it is NP-hard to decide whether [math] admits an obstacle representation using [math] as the only obstacle.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"48 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146395","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 2, Page 1526-1536, June 2024. Abstract. Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence from linear independence, deep connections between these two areas are found and are still undergoing active research. In this paper, we give a characterization of the embedding of matroids into conditional independence structures and its oriented counterpart, which leads to new axiom systems of matroids and oriented matroids.
{"title":"An Axiomatization of Matroids and Oriented Matroids as Conditional Independence Models","authors":"Xiangying Chen","doi":"10.1137/23m1558653","DOIUrl":"https://doi.org/10.1137/23m1558653","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1526-1536, June 2024. <br/> Abstract. Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence from linear independence, deep connections between these two areas are found and are still undergoing active research. In this paper, we give a characterization of the embedding of matroids into conditional independence structures and its oriented counterpart, which leads to new axiom systems of matroids and oriented matroids.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"43 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062659","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 2, Page 1492-1525, June 2024. Abstract. We describe a robust methodology, based on the martingale argument of Nachmias and Peres and random walk estimates, to obtain simple upper and lower bounds on the size of a maximal component in several random graphs at criticality. Even though the main result is not new, we believe the material presented here is interesting because it unifies several proofs found in the literature into a common framework. More specifically, we give easy-to-check conditions that, when satisfied, allow an immediate derivation of the above-mentioned bounds.
{"title":"A Simple Path to Component Sizes in Critical Random Graphs","authors":"Umberto De Ambroggio","doi":"10.1137/22m151056x","DOIUrl":"https://doi.org/10.1137/22m151056x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1492-1525, June 2024. <br/> Abstract. We describe a robust methodology, based on the martingale argument of Nachmias and Peres and random walk estimates, to obtain simple upper and lower bounds on the size of a maximal component in several random graphs at criticality. Even though the main result is not new, we believe the material presented here is interesting because it unifies several proofs found in the literature into a common framework. More specifically, we give easy-to-check conditions that, when satisfied, allow an immediate derivation of the above-mentioned bounds.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889887","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 2, Page 1462-1471, June 2024. Abstract. We show that for every [math], the maximal running time of the [math]-bootstrap percolation in the complete [math]-uniform hypergraph on [math] vertices [math] is [math]. This answers a recent question of Noel and Ranganathan in the affirmative and disproves a conjecture of theirs. Moreover, we show that the prefactor is of the form [math] as [math].
{"title":"The Maximal Running Time of Hypergraph Bootstrap Percolation","authors":"Ivailo Hartarsky, Lyuben Lichev","doi":"10.1137/22m151995x","DOIUrl":"https://doi.org/10.1137/22m151995x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1462-1471, June 2024. <br/> Abstract. We show that for every [math], the maximal running time of the [math]-bootstrap percolation in the complete [math]-uniform hypergraph on [math] vertices [math] is [math]. This answers a recent question of Noel and Ranganathan in the affirmative and disproves a conjecture of theirs. Moreover, we show that the prefactor is of the form [math] as [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"32 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883085","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 2, Page 1472-1491, June 2024. Abstract. Recently, it was proved by Bérczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were left open. We first show that the problem remains hard if the matroid is graphic, answering a question of Bérczi and Schwarcz. As another special case, we consider the problem of deciding whether a given digraph can be factorized into subgraphs which are spanning trees in the underlying sense and respect upper bounds on the indegree of every vertex. We prove that this problem is also hard. This answers a question of Frank. In the second part of the article, we deal with the relaxed problem of covering the ground set of a matroid by rainbow bases. Among other results, we show that there is a linear function [math] such that every matroid that can be factorized into [math] bases for some [math] can be covered by [math] rainbow bases if every partition class contains at most 2 elements.
{"title":"Rainbow Bases in Matroids","authors":"Florian Hörsch, Tomáš Kaiser, Matthias Kriesell","doi":"10.1137/22m1516750","DOIUrl":"https://doi.org/10.1137/22m1516750","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1472-1491, June 2024. <br/> Abstract. Recently, it was proved by Bérczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were left open. We first show that the problem remains hard if the matroid is graphic, answering a question of Bérczi and Schwarcz. As another special case, we consider the problem of deciding whether a given digraph can be factorized into subgraphs which are spanning trees in the underlying sense and respect upper bounds on the indegree of every vertex. We prove that this problem is also hard. This answers a question of Frank. In the second part of the article, we deal with the relaxed problem of covering the ground set of a matroid by rainbow bases. Among other results, we show that there is a linear function [math] such that every matroid that can be factorized into [math] bases for some [math] can be covered by [math] rainbow bases if every partition class contains at most 2 elements.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883083","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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