SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1250-1268, June 2024. Abstract. We completely characterize the triangulations of the projective plane that admit a spanning bipartite quadrangulation subgraph. This is an affirmative answer to a question by Kündgen and Ramamurthi [J. Combin. Theory Ser. B, 85 (2002), pp. 307–337] for the projective planar case.
{"title":"Spanning Bipartite Quadrangulations of Triangulations of the Projective Plane","authors":"Kenta Noguchi","doi":"10.1137/23m1566960","DOIUrl":"https://doi.org/10.1137/23m1566960","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1250-1268, June 2024. <br/> Abstract. We completely characterize the triangulations of the projective plane that admit a spanning bipartite quadrangulation subgraph. This is an affirmative answer to a question by Kündgen and Ramamurthi [J. Combin. Theory Ser. B, 85 (2002), pp. 307–337] for the projective planar case.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"13 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563143","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}
Natalie Behague, Tom Johnston, Shoham Letzter, Natasha Morrison, Shannon Ogden
SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1239-1249, June 2024. Abstract. Given a graph [math], we say that an edge-colored graph [math] is [math]-rainbow saturated if it does not contain a rainbow copy of [math], but the addition of any nonedge in any color creates a rainbow copy of [math]. The rainbow saturation number [math] is the minimum number of edges among all [math]-rainbow saturated edge-colored graphs on [math] vertices. We prove that for any nonempty graph [math], the rainbow saturation number is linear in [math], thus proving a conjecture of Girão, Lewis, and Popielarz. In addition, we give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Girão, Lewis, and Popielarz.
{"title":"The Rainbow Saturation Number Is Linear","authors":"Natalie Behague, Tom Johnston, Shoham Letzter, Natasha Morrison, Shannon Ogden","doi":"10.1137/23m1566881","DOIUrl":"https://doi.org/10.1137/23m1566881","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1239-1249, June 2024. <br/> Abstract. Given a graph [math], we say that an edge-colored graph [math] is [math]-rainbow saturated if it does not contain a rainbow copy of [math], but the addition of any nonedge in any color creates a rainbow copy of [math]. The rainbow saturation number [math] is the minimum number of edges among all [math]-rainbow saturated edge-colored graphs on [math] vertices. We prove that for any nonempty graph [math], the rainbow saturation number is linear in [math], thus proving a conjecture of Girão, Lewis, and Popielarz. In addition, we give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Girão, Lewis, and Popielarz.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"47 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140562971","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 1222-1238, June 2024. Abstract. The determinant lower bound of Lovász, Spencer, and Vesztergombi [European J. Combin., 7 (1986), pp. 151–160] is a general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovász, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the determinant lower bound. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of [math] subsets of a universe of size [math] is on the order of [math]. On the other hand, building upon work of Matoušek [Proc. Amer. Math. Soc., 141 (2013), pp. 451–460], Jiang and Reis [in Proceedings of the Symposium on Simplicity in Algorithms (SOSA), SIAM, Philadelphia, 2022, pp. 308–313] showed that this gap is always bounded up to constants by [math]. This is tight when [math] is polynomial in [math] but leaves open the case of large [math]. We show that the bound of Jiang and Reis is tight for nearly the entire range of [math]. Our proof amplifies the discrepancy lower bounds of a set system derived from the discrete Haar basis via Kronecker products.
SIAM 离散数学杂志》第 38 卷第 2 期第 1222-1238 页，2024 年 6 月。 摘要。Lovász、Spencer 和 Vesztergombi 的行列式下界 [European J. Combin., 7 (1986), pp.在他们的论文中，Lovász、Spencer 和 Vesztergombi 询问遗传差异是否也可以通过行列式下界的函数从上而下地限定。霍夫曼对此的回答是否定的，对于大小为[math]的宇宙的[math]子集的集合系统，这两个量之间已知的最大乘法差距是[math]数量级。另一方面，在马图塞克 [Proc. Amer. Math. Soc., 141 (2013), pp. 451-460] 的工作基础上，蒋和雷斯 [in Proceedings of the Symposium on Simplicity in Algorithms (SOSA), SIAM, Philadelphia, 2022, pp.当 [math] 是 [math] 的多项式时，这个界限很窄，但当 [math] 较大时，这个界限就很宽了。我们证明，几乎在 [math] 的整个范围内，Jiang 和 Reis 的约束都很紧。我们的证明扩大了通过克朗内克积从离散哈尔基导出的集合系统的差异下界。
{"title":"On the Gap Between Hereditary Discrepancy and the Determinant Lower Bound","authors":"Lily Li, Aleksandar Nikolov","doi":"10.1137/23m1566790","DOIUrl":"https://doi.org/10.1137/23m1566790","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1222-1238, June 2024. <br/> Abstract. The determinant lower bound of Lovász, Spencer, and Vesztergombi [European J. Combin., 7 (1986), pp. 151–160] is a general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovász, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the determinant lower bound. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of [math] subsets of a universe of size [math] is on the order of [math]. On the other hand, building upon work of Matoušek [Proc. Amer. Math. Soc., 141 (2013), pp. 451–460], Jiang and Reis [in Proceedings of the Symposium on Simplicity in Algorithms (SOSA), SIAM, Philadelphia, 2022, pp. 308–313] showed that this gap is always bounded up to constants by [math]. This is tight when [math] is polynomial in [math] but leaves open the case of large [math]. We show that the bound of Jiang and Reis is tight for nearly the entire range of [math]. Our proof amplifies the discrepancy lower bounds of a set system derived from the discrete Haar basis via Kronecker products.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"242 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563420","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 1202-1221, June 2024. Abstract. Let [math] be a bipartite graph where every node has a strict ranking of its neighbors. For any node, its preferences over neighbors extend naturally to preferences over matchings. A maximum matching [math] in [math] is a popular max-matching if there is no maximum matching more popular than [math]. In other words, for any maximum matching [math], the number of nodes that prefer [math] to [math] is at least the number of nodes that prefer [math] to [math]. It is known that popular max-matchings always exist in [math] and one such matching can be efficiently computed. In this paper we are in the weighted setting, i.e., there is a cost function [math], and our goal is to find a min-cost popular max-matching. We prove that such a matching can be computed in polynomial time by showing a compact extended formulation for the popular max-matching polytope. By contrast, it is known that the popular matching polytope has near-exponential extension complexity and finding a min-cost popular matching is NP-hard. We also consider Pareto-optimality. Though it is easy to find a Pareto-optimal matching/max-matching, we show that it is NP-hard to find a min-cost Pareto-optimal matching/max-matching.
{"title":"Maximum Matchings and Popularity","authors":"Telikepalli Kavitha","doi":"10.1137/22m1523248","DOIUrl":"https://doi.org/10.1137/22m1523248","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1202-1221, June 2024. <br/> Abstract. Let [math] be a bipartite graph where every node has a strict ranking of its neighbors. For any node, its preferences over neighbors extend naturally to preferences over matchings. A maximum matching [math] in [math] is a popular max-matching if there is no maximum matching more popular than [math]. In other words, for any maximum matching [math], the number of nodes that prefer [math] to [math] is at least the number of nodes that prefer [math] to [math]. It is known that popular max-matchings always exist in [math] and one such matching can be efficiently computed. In this paper we are in the weighted setting, i.e., there is a cost function [math], and our goal is to find a min-cost popular max-matching. We prove that such a matching can be computed in polynomial time by showing a compact extended formulation for the popular max-matching polytope. By contrast, it is known that the popular matching polytope has near-exponential extension complexity and finding a min-cost popular matching is NP-hard. We also consider Pareto-optimality. Though it is easy to find a Pareto-optimal matching/max-matching, we show that it is NP-hard to find a min-cost Pareto-optimal matching/max-matching.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"39 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563145","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 1193-1201, June 2024. Abstract. Given a simple Eulerian binary matroid [math], what is the minimum number of disjoint circuits necessary to decompose [math]? We prove that [math] many circuits suffice if [math] is the complete binary matroid, for certain values of [math], and that [math] many circuits suffice for general [math]. We also determine the asymptotic behavior of the minimum number of circuits in an odd-cover of [math].
{"title":"Circuit Decompositions of Binary Matroids","authors":"Bryce Frederickson, Lukas Michel","doi":"10.1137/23m1587439","DOIUrl":"https://doi.org/10.1137/23m1587439","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1193-1201, June 2024. <br/> Abstract. Given a simple Eulerian binary matroid [math], what is the minimum number of disjoint circuits necessary to decompose [math]? We prove that [math] many circuits suffice if [math] is the complete binary matroid, for certain values of [math], and that [math] many circuits suffice for general [math]. We also determine the asymptotic behavior of the minimum number of circuits in an odd-cover of [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"109 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563510","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 1, Page 1191-1192, March 2024. Abstract. We spotted an error in our publication More applications of the d-neighbor equivalence: Acyclicity and Connetivity constraints [SIAM J. Discrete Math., 35 (2021), pp. 1881–1926]. We explain the problem and suggest a simple correction.
{"title":"Erratum: More Applications of the [math]-Neighbor Equivalence: Acyclicity and Connectivity Constraints","authors":"Benjamin Bergougnoux, Mamadou M. Kanté","doi":"10.1137/23m157644x","DOIUrl":"https://doi.org/10.1137/23m157644x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1191-1192, March 2024. <br/> Abstract. We spotted an error in our publication More applications of the d-neighbor equivalence: Acyclicity and Connetivity constraints [SIAM J. Discrete Math., 35 (2021), pp. 1881–1926]. We explain the problem and suggest a simple correction.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"97 4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299019","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 1, Page 1158-1190, March 2024. Abstract. The List-3-Coloring Problem is to decide, given a graph [math] and a list [math] of colors assigned to each vertex [math] of [math], whether [math] admits a proper coloring [math] with [math] for every vertex [math] of [math], and the 3-Coloring Problem is the List-3-Coloring Problem on instances with [math] for every vertex [math] of [math]. The List-3-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to [math]-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph [math]), it remains NP-complete unless [math] is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the 3-Coloring Problem on [math]-free graphs (where [math] denotes the eight-vertex path) is unknown. Here we consider a variant of the List-3-Coloring Problem called the Ordered Graph List-3-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs [math] and [math], we say [math] is [math]-free if [math] is not isomorphic to an induced subgraph of [math] with the isomorphism preserving the linear order. We prove, assuming [math] to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-3-Coloring Problem restricted to [math]-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if [math] has at most one edge, and remains NP-complete if [math] has at least three edges. Moreover, in the case where [math] has exactly two edges, we give a complete dichotomy when the two edges of [math] share an end, and prove several NP-completeness results when the two edges of [math] do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs.
{"title":"List-3-Coloring Ordered Graphs with a Forbidden Induced Subgraphs","authors":"Sepehr Hajebi, Yanjia Li, Sophie Spirkl","doi":"10.1137/22m1515768","DOIUrl":"https://doi.org/10.1137/22m1515768","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1158-1190, March 2024. <br/> Abstract. The List-3-Coloring Problem is to decide, given a graph [math] and a list [math] of colors assigned to each vertex [math] of [math], whether [math] admits a proper coloring [math] with [math] for every vertex [math] of [math], and the 3-Coloring Problem is the List-3-Coloring Problem on instances with [math] for every vertex [math] of [math]. The List-3-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to [math]-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph [math]), it remains NP-complete unless [math] is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the 3-Coloring Problem on [math]-free graphs (where [math] denotes the eight-vertex path) is unknown. Here we consider a variant of the List-3-Coloring Problem called the Ordered Graph List-3-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs [math] and [math], we say [math] is [math]-free if [math] is not isomorphic to an induced subgraph of [math] with the isomorphism preserving the linear order. We prove, assuming [math] to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-3-Coloring Problem restricted to [math]-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if [math] has at most one edge, and remains NP-complete if [math] has at least three edges. Moreover, in the case where [math] has exactly two edges, we give a complete dichotomy when the two edges of [math] share an end, and prove several NP-completeness results when the two edges of [math] do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140205504","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 1, Page 1113-1157, March 2024. Abstract. In the infinite regular tree [math] with [math], we consider families [math], indexed by vertices [math] and nonnegative integers (“discrete time steps”) [math], of probability measures such that [math] if the distances [math] and [math] are equal. Let [math] be a positive integer, and let [math] and [math] be two vertices in the tree which are at distance [math] apart. We compute a formula for the transportation distance [math] in terms of generating functions. In the special case where [math] are measures from simple random walks after [math] time steps, we establish the linear asymptotic formula [math], as [math], and give the formulas for the coefficients [math] and [math] in closed forms. We also obtain linear asymptotic formulas when [math] is the uniform distribution on the sphere or on the ball of radius [math] as [math]. We show that these six coefficients (two from the simple random walk, two from the uniform distribution on the sphere, and two from the uniform distribution on the ball) are related by inequalities.
{"title":"Transportation Distance between Probability Measures on the Infinite Regular Tree","authors":"Pakawut Jiradilok, Supanat Kamtue","doi":"10.1137/21m1448781","DOIUrl":"https://doi.org/10.1137/21m1448781","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1113-1157, March 2024. <br/> Abstract. In the infinite regular tree [math] with [math], we consider families [math], indexed by vertices [math] and nonnegative integers (“discrete time steps”) [math], of probability measures such that [math] if the distances [math] and [math] are equal. Let [math] be a positive integer, and let [math] and [math] be two vertices in the tree which are at distance [math] apart. We compute a formula for the transportation distance [math] in terms of generating functions. In the special case where [math] are measures from simple random walks after [math] time steps, we establish the linear asymptotic formula [math], as [math], and give the formulas for the coefficients [math] and [math] in closed forms. We also obtain linear asymptotic formulas when [math] is the uniform distribution on the sphere or on the ball of radius [math] as [math]. We show that these six coefficients (two from the simple random walk, two from the uniform distribution on the sphere, and two from the uniform distribution on the ball) are related by inequalities.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151871","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}
Debsoumya Chakraborti, Kevin Hendrey, Ben Lund, Casey Tompkins
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1090-1112, March 2024. Abstract. We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph [math] is called [math]-rainbow saturated if [math] does not contain a rainbow copy of [math] and adding an edge of any color to [math] creates a rainbow copy of [math]. The rainbow saturation number [math] is the minimum number of edges in an [math]-vertex [math]-rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that [math] for fixed [math]. Disproving this conjecture, we establish that for every [math], there exists a constant [math] such that [math] and [math]. Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number and asked whether this is equal to the rainbow saturation number of [math], since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant [math] resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored [math]-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are [math]-saturated with respect to the operation of deleting one edge and adding two edges.
{"title":"Rainbow Saturation for Complete Graphs","authors":"Debsoumya Chakraborti, Kevin Hendrey, Ben Lund, Casey Tompkins","doi":"10.1137/23m1565875","DOIUrl":"https://doi.org/10.1137/23m1565875","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1090-1112, March 2024. <br/> Abstract. We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph [math] is called [math]-rainbow saturated if [math] does not contain a rainbow copy of [math] and adding an edge of any color to [math] creates a rainbow copy of [math]. The rainbow saturation number [math] is the minimum number of edges in an [math]-vertex [math]-rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that [math] for fixed [math]. Disproving this conjecture, we establish that for every [math], there exists a constant [math] such that [math] and [math]. Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number and asked whether this is equal to the rainbow saturation number of [math], since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant [math] resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored [math]-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are [math]-saturated with respect to the operation of deleting one edge and adding two edges.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151873","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 1, Page 1057-1089, March 2024. Abstract. The planar graph product structure theorem of Dujmović et al. [J. ACM, 67 (2020), 22] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colorings, centered colorings, and adjacency labeling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond-planar graph classes. The key observation that drives our work is that many beyond-planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that powers of bounded degree planar graphs, [math]-planar, [math]-cluster planar, fan-planar, and [math]-fan-bundle planar graphs have such a shallow-minor structure. Using a combination of old and new results, we deduce that these classes have bounded queue-number, bounded nonrepetitive chromatic number, polynomial [math]-centered chromatic numbers, linear strong coloring numbers, and cubic weak coloring numbers. In addition, we show that [math]-gap planar graphs have at least exponential local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.
{"title":"Shallow Minors, Graph Products, and Beyond-Planar Graphs","authors":"Robert Hickingbotham, David R. Wood","doi":"10.1137/22m1540296","DOIUrl":"https://doi.org/10.1137/22m1540296","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1057-1089, March 2024. <br/> Abstract. The planar graph product structure theorem of Dujmović et al. [J. ACM, 67 (2020), 22] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colorings, centered colorings, and adjacency labeling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond-planar graph classes. The key observation that drives our work is that many beyond-planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that powers of bounded degree planar graphs, [math]-planar, [math]-cluster planar, fan-planar, and [math]-fan-bundle planar graphs have such a shallow-minor structure. Using a combination of old and new results, we deduce that these classes have bounded queue-number, bounded nonrepetitive chromatic number, polynomial [math]-centered chromatic numbers, linear strong coloring numbers, and cubic weak coloring numbers. In addition, we show that [math]-gap planar graphs have at least exponential local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125187","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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