Pierre Aboulker, Thomas Bellitto, F. Havet, Clément Rambaud
{"title":"On the Minimum Number of Arcs in (boldsymbol{k})-Dicritical Oriented Graphs","authors":"Pierre Aboulker, Thomas Bellitto, F. Havet, Clément Rambaud","doi":"10.1137/23m1553753","DOIUrl":"https://doi.org/10.1137/23m1553753","url":null,"abstract":"","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141349838","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}
Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1840-1862, June 2024. Abstract. We show that if the edges or vertices of an undirected graph [math] can be covered by [math] shortest paths, then the pathwidth of [math] is upper-bounded by a single-exponential function of [math]. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph [math] and a set of [math] pairs of vertices called terminals, asks whether [math] can be covered by [math] shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph [math] and a set of [math] terminals, asks whether there exist [math] shortest paths covering [math], each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter [math].
{"title":"On Graphs Coverable by [math] Shortest Paths","authors":"Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca","doi":"10.1137/23m1564511","DOIUrl":"https://doi.org/10.1137/23m1564511","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1840-1862, June 2024. <br/> Abstract. We show that if the edges or vertices of an undirected graph [math] can be covered by [math] shortest paths, then the pathwidth of [math] is upper-bounded by a single-exponential function of [math]. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph [math] and a set of [math] pairs of vertices called terminals, asks whether [math] can be covered by [math] shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph [math] and a set of [math] terminals, asks whether there exist [math] shortest paths covering [math], each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508466","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}
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":null,"pages":null},"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":null,"pages":null},"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}
{"title":"A Stability Result of the Pósa Lemma","authors":"Jie Ma, Long-Tu Yuan","doi":"10.1137/20m1382143","DOIUrl":"https://doi.org/10.1137/20m1382143","url":null,"abstract":"","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141379040","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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
{"title":"Concepts of Dimension for Convex Geometries","authors":"Kolja Knauer, William T. Trotter","doi":"10.1137/23m1559853","DOIUrl":"https://doi.org/10.1137/23m1559853","url":null,"abstract":"","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141111212","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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