Natalie Behague, Natasha Morrison, Jonathan A. Noel
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2335-2360, September 2024. Abstract. A graph [math] is common if the limit as [math] of the minimum density of monochromatic labeled copies of [math] in an edge coloring of [math] with red and blue is attained by a sequence of quasirandom colorings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair [math] of such graphs, there exists [math] such that an appropriate linear combination of red copies of [math] and blue copies of [math] is minimized by a quasirandom coloring in which [math] edges are red; such a pair [math] is said to be [math]-common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a [math]-common pair [math] such that [math] is uncommon.
{"title":"Off-Diagonal Commonality of Graphs via Entropy","authors":"Natalie Behague, Natasha Morrison, Jonathan A. Noel","doi":"10.1137/23m1625342","DOIUrl":"https://doi.org/10.1137/23m1625342","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2335-2360, September 2024. <br/> Abstract. A graph [math] is common if the limit as [math] of the minimum density of monochromatic labeled copies of [math] in an edge coloring of [math] with red and blue is attained by a sequence of quasirandom colorings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair [math] of such graphs, there exists [math] such that an appropriate linear combination of red copies of [math] and blue copies of [math] is minimized by a quasirandom coloring in which [math] edges are red; such a pair [math] is said to be [math]-common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a [math]-common pair [math] such that [math] is uncommon.","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":"142202431","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}
József Balogh, Anita Liebenau, Letícia Mattos, Natasha Morrison
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2297-2311, September 2024. Abstract. We address a problem which is a generalization of Turán-type problems recently introduced by Imolay, Karl, Nagy, and Váli. Let [math] be a fixed graph and let [math] be the union of [math] edge-disjoint copies of [math], namely [math], where each [math] is isomorphic to a fixed graph [math] and [math] for all [math]. We call a subgraph [math] multicolored if [math] and [math] share at most one edge for all [math]. Define [math] to be the maximum value [math] such that there exists [math] on [math] vertices without a multicolored copy of [math]. We show that [math] and that all extremal graphs are close to a blow-up of the 5-cycle. This bound is tight up to the linear error term.
{"title":"On Multicolor Turán Numbers","authors":"József Balogh, Anita Liebenau, Letícia Mattos, Natasha Morrison","doi":"10.1137/24m1639488","DOIUrl":"https://doi.org/10.1137/24m1639488","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2297-2311, September 2024. <br/> Abstract. We address a problem which is a generalization of Turán-type problems recently introduced by Imolay, Karl, Nagy, and Váli. Let [math] be a fixed graph and let [math] be the union of [math] edge-disjoint copies of [math], namely [math], where each [math] is isomorphic to a fixed graph [math] and [math] for all [math]. We call a subgraph [math] multicolored if [math] and [math] share at most one edge for all [math]. Define [math] to be the maximum value [math] such that there exists [math] on [math] vertices without a multicolored copy of [math]. We show that [math] and that all extremal graphs are close to a blow-up of the 5-cycle. This bound is tight up to the linear error term.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202433","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 2289-2296, September 2024. Abstract. We provide a combinatorial and self-contained proof of a result following from G. Besson [Ann. Inst. Fourier, 30 (1980), pp. 109–128] and Y. Colin de Verdière [Ann. Sci. Éc. Norm. Supér., 20 (1987), pp. 599–615] that for all graphs [math] embedded on a surface [math], the Colin de Verdière parameter [math] is upper bounded by [math].
SIAM 离散数学杂志》，第 38 卷第 3 期，第 2289-2296 页，2024 年 9 月。 摘要。我们对贝松 (G. Besson) [Ann. Inst. Fourier, 30 (1980), pp.
{"title":"A Linear Bound for the Colin de Verdière Parameter [math] for Graphs Embedded on Surfaces","authors":"Camille Lanuel, Francis Lazarus, Rudi Pendavingh","doi":"10.1137/23m1623628","DOIUrl":"https://doi.org/10.1137/23m1623628","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2289-2296, September 2024. <br/> Abstract. We provide a combinatorial and self-contained proof of a result following from G. Besson [Ann. Inst. Fourier, 30 (1980), pp. 109–128] and Y. Colin de Verdière [Ann. Sci. Éc. Norm. Supér., 20 (1987), pp. 599–615] that for all graphs [math] embedded on a surface [math], the Colin de Verdière parameter [math] is upper bounded by [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202435","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 2260-2288, September 2024. Abstract. A celebrated conjecture of Tuza states that in any finite graph the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. For an [math]-uniform hypergraph ([math]-graph) [math], let [math] be the minimum size of a cover of edges by [math]-sets of vertices, and let [math] be the maximum size of a set of edges pairwise intersecting in fewer than [math] vertices. Aharoni and Zerbib proposed the following generalization of Tuza’s conjecture: For any [math]-graph [math], [math]. Let [math] be the uniformly random [math]-graph on [math] vertices. We show that for [math] and any [math], [math] satisfies the Aharoni–Zerbib conjecture with high probability (w.h.p.), i.e., with probability approaching 1 as [math]. We also show that there is a [math] such that for any [math] and any [math], [math] w.h.p. Furthermore, we may take [math], for any [math], by restricting to sufficiently large [math] (depending on [math]).
{"title":"Generalized Tuza’s Conjecture for Random Hypergraphs","authors":"Abdul Basit, David Galvin","doi":"10.1137/23m1587014","DOIUrl":"https://doi.org/10.1137/23m1587014","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2260-2288, September 2024. <br/> Abstract. A celebrated conjecture of Tuza states that in any finite graph the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. For an [math]-uniform hypergraph ([math]-graph) [math], let [math] be the minimum size of a cover of edges by [math]-sets of vertices, and let [math] be the maximum size of a set of edges pairwise intersecting in fewer than [math] vertices. Aharoni and Zerbib proposed the following generalization of Tuza’s conjecture: For any [math]-graph [math], [math]. Let [math] be the uniformly random [math]-graph on [math] vertices. We show that for [math] and any [math], [math] satisfies the Aharoni–Zerbib conjecture with high probability (w.h.p.), i.e., with probability approaching 1 as [math]. We also show that there is a [math] such that for any [math] and any [math], [math] w.h.p. Furthermore, we may take [math], for any [math], by restricting to sufficiently large [math] (depending on [math]).","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141864261","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 2243-2259, September 2024. Abstract. Given [math] points [math] on the unit circle in [math] and a number [math], we investigate the minimizers of the functional [math]. While it is known that each of these minimizers is a spanning set for [math], less is known about their number as a function of [math] and [math] especially for relatively small [math]. In this paper we show that there is unique minimum for this functional for all [math] and all odd [math]. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for [math] odd, there exists a sequence of points [math] so that a unique (up to some isometries) minimizer exists on each of the subintervals [math].
{"title":"Phase Transitions for the Minimizers of the [math]-Frame Potentials in [math]","authors":"Radel Ben-Av, Xuemei Chen, Assaf Goldberger, Shujie Kang, Kasso A. Okoudjou","doi":"10.1137/22m1539915","DOIUrl":"https://doi.org/10.1137/22m1539915","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2243-2259, September 2024. <br/> Abstract. Given [math] points [math] on the unit circle in [math] and a number [math], we investigate the minimizers of the functional [math]. While it is known that each of these minimizers is a spanning set for [math], less is known about their number as a function of [math] and [math] especially for relatively small [math]. In this paper we show that there is unique minimum for this functional for all [math] and all odd [math]. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for [math] odd, there exists a sequence of points [math] so that a unique (up to some isometries) minimizer exists on each of the subintervals [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771395","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 2226-2242, September 2024. Abstract. We consider the problem of finding a maximum popular matching in a many-to-many matching setting with two-sided preferences and matroid constraints. This problem was proposed by Kamiyama [Theoret. Comput. Sci., 809 (2020), pp. 265–276] and solved in the special case where matroids are base orderable. Utilizing a newly shown matroid exchange property, we show that the problem is tractable for arbitrary matroids. We further investigate a different notion of popularity, where the agents vote with respect to lexicographic preferences, and show that both existence and verification problems become coNP-hard even in the [math]-matching case.
{"title":"Solving the Maximum Popular Matching Problem with Matroid Constraints","authors":"Gergely Csáji, Tamás Király, Yu Yokoi","doi":"10.1137/23m1579911","DOIUrl":"https://doi.org/10.1137/23m1579911","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2226-2242, September 2024. <br/> Abstract. We consider the problem of finding a maximum popular matching in a many-to-many matching setting with two-sided preferences and matroid constraints. This problem was proposed by Kamiyama [Theoret. Comput. Sci., 809 (2020), pp. 265–276] and solved in the special case where matroids are base orderable. Utilizing a newly shown matroid exchange property, we show that the problem is tractable for arbitrary matroids. We further investigate a different notion of popularity, where the agents vote with respect to lexicographic preferences, and show that both existence and verification problems become coNP-hard even in the [math]-matching case.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771396","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}
Elena S. Hafner, Karola Mészáros, Linus Setiabrata, Avery St. Dizier
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2194-2225, September 2024. Abstract. We introduce bubbling diagrams and show that they compute the support of the Grothendieck polynomial of any vexillary permutation. Using these diagrams, we show that the support of the top homogeneous component of such a Grothendieck polynomial coincides with the support of the dual character of an explicit flagged Weyl module. We also show that the homogenized Grothendieck polynomial of a vexillary permutation has M-convex support.
{"title":"M-Convexity of Vexillary Grothendieck Polynomials via Bubbling","authors":"Elena S. Hafner, Karola Mészáros, Linus Setiabrata, Avery St. Dizier","doi":"10.1137/23m1599082","DOIUrl":"https://doi.org/10.1137/23m1599082","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2194-2225, September 2024. <br/> Abstract. We introduce bubbling diagrams and show that they compute the support of the Grothendieck polynomial of any vexillary permutation. Using these diagrams, we show that the support of the top homogeneous component of such a Grothendieck polynomial coincides with the support of the dual character of an explicit flagged Weyl module. We also show that the homogenized Grothendieck polynomial of a vexillary permutation has M-convex support.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771397","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 2181-2193, September 2024. Abstract. For any hereditary graph class [math], we construct optimal adjacency labeling schemes for the classes of subgraphs and induced subgraphs of Cartesian products of graphs in [math]. As a consequence, we show that if [math] admits efficient adjacency labels (or, equivalently, small induced-universal graphs) meeting the information-theoretic minimum, then so do the classes of subgraphs and induced subgraphs of Cartesian products of graphs in [math]. Our proof uses ideas from randomized communication complexity, hashing, and additive combinatorics and improves upon recent results of Chepoi, Labourel, and Ratel [J. Graph Theory, 93 (2020), pp. 64–87].
{"title":"Optimal Adjacency Labels for Subgraphs of Cartesian Products","authors":"Louis Esperet, Nathaniel Harms, Viktor Zamaraev","doi":"10.1137/23m1587713","DOIUrl":"https://doi.org/10.1137/23m1587713","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2181-2193, September 2024. <br/> Abstract. For any hereditary graph class [math], we construct optimal adjacency labeling schemes for the classes of subgraphs and induced subgraphs of Cartesian products of graphs in [math]. As a consequence, we show that if [math] admits efficient adjacency labels (or, equivalently, small induced-universal graphs) meeting the information-theoretic minimum, then so do the classes of subgraphs and induced subgraphs of Cartesian products of graphs in [math]. Our proof uses ideas from randomized communication complexity, hashing, and additive combinatorics and improves upon recent results of Chepoi, Labourel, and Ratel [J. Graph Theory, 93 (2020), pp. 64–87].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745427","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 2108-2131, September 2024. Abstract. We study the computational efficiency of approaches, based on Hilbert’s Nullstellensatz, which use systems of linear equations for detecting noncolorability of graphs having large girth and chromatic number. We show that for every non-[math]-colorable graph with [math] vertices and girth [math], the algorithm is required to solve systems of size at least [math] in order to detect its non-[math]-colorability.
{"title":"Graphs with Large Girth and Chromatic Number are Hard for Nullstellensatz","authors":"Julian Romero, Levent Tunçel","doi":"10.1137/23m1553273","DOIUrl":"https://doi.org/10.1137/23m1553273","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2108-2131, September 2024. <br/> Abstract. We study the computational efficiency of approaches, based on Hilbert’s Nullstellensatz, which use systems of linear equations for detecting noncolorability of graphs having large girth and chromatic number. We show that for every non-[math]-colorable graph with [math] vertices and girth [math], the algorithm is required to solve systems of size at least [math] in order to detect its non-[math]-colorability.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609410","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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