SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 453-484, March 2024. Abstract. We present an isomorphism test for graphs of Euler genus [math] running in time [math]. Our algorithm provides the first explicit upper bound on the dependence on [math] for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time [math] for some function [math] (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude [math] as a minor. For such graphs, no fpt isomorphism test was known before. The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, our algorithm relies on the notion of [math]-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler–Leman algorithm. This concept may be of independent interest.
{"title":"Isomorphism Testing Parameterized by Genus and Beyond","authors":"Daniel Neuen","doi":"10.1137/22m1514076","DOIUrl":"https://doi.org/10.1137/22m1514076","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 453-484, March 2024. <br/> Abstract. We present an isomorphism test for graphs of Euler genus [math] running in time [math]. Our algorithm provides the first explicit upper bound on the dependence on [math] for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time [math] for some function [math] (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude [math] as a minor. For such graphs, no fpt isomorphism test was known before. The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, our algorithm relies on the notion of [math]-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler–Leman algorithm. This concept may be of independent interest.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139555109","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 380-411, March 2024. Abstract. Phylogenetic trees are used to model evolution: leaves are labeled to represent contemporary species (“taxa”), and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state form a connected subtree. Kelk and Stamoulis [Adv. Appl. Math., 84 (2017), pp. 34–46] showed how to efficiently count, list, and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterized algorithms can be used to speed up exponential-time algorithms for the maximum agreement forest problem in phylogenetics. Second, we revisit the quantity [math], defined as the number of convex characters on [math] in which each state appears on at least 2 taxa. We use this to give an algorithm with running time [math], where [math] is the golden ratio and [math] is the number of taxa in the input trees for computation of maximum parsimony distance on two state characters. By further restricting the characters counted by [math] we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to [math] and obtain a number of new results in themselves relevant to enumeration of matchings on at most binary trees.
{"title":"Convex Characters, Algorithms, and Matchings","authors":"Steven Kelk, Ruben Meuwese, Stephan Wagner","doi":"10.1137/21m1463999","DOIUrl":"https://doi.org/10.1137/21m1463999","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 380-411, March 2024. <br/> Abstract. Phylogenetic trees are used to model evolution: leaves are labeled to represent contemporary species (“taxa”), and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state form a connected subtree. Kelk and Stamoulis [Adv. Appl. Math., 84 (2017), pp. 34–46] showed how to efficiently count, list, and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterized algorithms can be used to speed up exponential-time algorithms for the maximum agreement forest problem in phylogenetics. Second, we revisit the quantity [math], defined as the number of convex characters on [math] in which each state appears on at least 2 taxa. We use this to give an algorithm with running time [math], where [math] is the golden ratio and [math] is the number of taxa in the input trees for computation of maximum parsimony distance on two state characters. By further restricting the characters counted by [math] we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to [math] and obtain a number of new results in themselves relevant to enumeration of matchings on at most binary trees.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500083","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 412-452, March 2024. Abstract. In the distributional Twenty Questions game, Bob chooses a number [math] from 1 to [math] according to a distribution [math], and Alice (who knows [math]) attempts to identify [math] using yes/no questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for [math], yet this strategy could potentially uses all [math] possible questions. Dagan et al. constructed a set of [math] questions which suffice to construct an optimal strategy for all [math], and showed that this number is optimal (up to subexponential factors) for infinitely many [math]. We determine the optimal size of such a set of questions for all [math] (up to subexponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the [math]-ary setting, obtaining similar results with 1.25 replaced by [math].
{"title":"Optimal Sets of Questions for Twenty Questions","authors":"Yuval Filmus, Idan Mehalel","doi":"10.1137/21m1424494","DOIUrl":"https://doi.org/10.1137/21m1424494","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 412-452, March 2024. <br/> Abstract. In the distributional Twenty Questions game, Bob chooses a number [math] from 1 to [math] according to a distribution [math], and Alice (who knows [math]) attempts to identify [math] using yes/no questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for [math], yet this strategy could potentially uses all [math] possible questions. Dagan et al. constructed a set of [math] questions which suffice to construct an optimal strategy for all [math], and showed that this number is optimal (up to subexponential factors) for infinitely many [math]. We determine the optimal size of such a set of questions for all [math] (up to subexponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the [math]-ary setting, obtaining similar results with 1.25 replaced by [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500167","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 348-379, March 2024. Abstract. We consider classes of objective functions of cardinality-constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of [math]-[math]-augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, [math]-augmentable functions, and weighted rank functions of an independence system of bounded rank quotient—as well as additional objective functions for which the greedy algorithm yields an approximation. For this general class of functions, we show a tight bound of [math] on the approximation ratio of the greedy algorithm that tightly interpolates between bounds from the literature for functions of bounded submodularity ratio and for [math]-augmentable functions. In particular, as a by-product, we close a gap in [A. Bernstein et al., Math. Program., 191 (2022), pp. 953–979] by obtaining a tight lower bound for [math]-augmentable functions for all [math]. For weighted rank functions of independence systems, our tight bound becomes [math], which recovers the known bound of [math] for independence systems of rank quotient at least [math].
{"title":"Unified Greedy Approximability beyond Submodular Maximization","authors":"Yann Disser, David Weckbecker","doi":"10.1137/22m1526952","DOIUrl":"https://doi.org/10.1137/22m1526952","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 348-379, March 2024. <br/> Abstract. We consider classes of objective functions of cardinality-constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of [math]-[math]-augmentable functions and prove that it encompasses several important subclasses, such as functions of bounded submodularity ratio, [math]-augmentable functions, and weighted rank functions of an independence system of bounded rank quotient—as well as additional objective functions for which the greedy algorithm yields an approximation. For this general class of functions, we show a tight bound of [math] on the approximation ratio of the greedy algorithm that tightly interpolates between bounds from the literature for functions of bounded submodularity ratio and for [math]-augmentable functions. In particular, as a by-product, we close a gap in [A. Bernstein et al., Math. Program., 191 (2022), pp. 953–979] by obtaining a tight lower bound for [math]-augmentable functions for all [math]. For weighted rank functions of independence systems, our tight bound becomes [math], which recovers the known bound of [math] for independence systems of rank quotient at least [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500095","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}
Noga Alon, Emil Powierski, Michael Savery, Alex Scott, Elizabeth Wilmer
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 327-347, March 2024. Abstract. For an oriented graph [math] and a set [math], the inversion of [math] in [math] is the digraph obtained by reversing the orientations of the edges of [math] with both endpoints in [math]. The inversion number of [math], [math], is the minimum number of inversions which can be applied in turn to [math] to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each [math] and tournament [math], the problem of deciding whether [math] is solvable in time [math], which is tight for all [math]. In particular, the problem is fixed-parameter tractable when parameterized by [math]. On the other hand, we build on their work to prove their conjecture that for [math] the problem of deciding whether a general oriented graph [math] has [math] is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called dijoin digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an [math]-vertex tournament is [math].
{"title":"Invertibility of Digraphs and Tournaments","authors":"Noga Alon, Emil Powierski, Michael Savery, Alex Scott, Elizabeth Wilmer","doi":"10.1137/23m1547135","DOIUrl":"https://doi.org/10.1137/23m1547135","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 327-347, March 2024. <br/> Abstract. For an oriented graph [math] and a set [math], the inversion of [math] in [math] is the digraph obtained by reversing the orientations of the edges of [math] with both endpoints in [math]. The inversion number of [math], [math], is the minimum number of inversions which can be applied in turn to [math] to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each [math] and tournament [math], the problem of deciding whether [math] is solvable in time [math], which is tight for all [math]. In particular, the problem is fixed-parameter tractable when parameterized by [math]. On the other hand, we build on their work to prove their conjecture that for [math] the problem of deciding whether a general oriented graph [math] has [math] is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called dijoin digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an [math]-vertex tournament is [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139500323","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 316-326, March 2024. Abstract. A graph [math] is called common and, respectively, strongly common if the number of monochromatic copies of [math] in a 2-edge-coloring [math] of a large clique is asymptotically minimized by the random coloring with an equal proportion of each color and, respectively, by the random coloring with the same proportion of each color as in [math]. A well-known theorem of Jagger, Št’ovíček, and Thomason states that every graph containing a [math] is not common. Here we prove an analogous result that every graph containing a [math] and with at least four edges is not strongly common.
{"title":"A Property on Monochromatic Copies of Graphs Containing a Triangle","authors":"Hao Chen, Jie Ma","doi":"10.1137/23m1564894","DOIUrl":"https://doi.org/10.1137/23m1564894","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 316-326, March 2024. <br/> Abstract. A graph [math] is called common and, respectively, strongly common if the number of monochromatic copies of [math] in a 2-edge-coloring [math] of a large clique is asymptotically minimized by the random coloring with an equal proportion of each color and, respectively, by the random coloring with the same proportion of each color as in [math]. A well-known theorem of Jagger, Št’ovíček, and Thomason states that every graph containing a [math] is not common. Here we prove an analogous result that every graph containing a [math] and with at least four edges is not strongly common.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139461450","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}
Davide Bilò, Tobias Friedrich, Pascal Lenzner, Anna Melnichenko
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 277-315, March 2024. Abstract. Network creation games are a well-known approach for explaining and analyzing the structure, quality, and dynamics of real-world networks that evolved via the interaction of selfish agents without a central authority. In these games selfish agents corresponding to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games only considered the creation of networks with unit-weight edges. In practice, e.g., when constructing a fiber-optic network, the choice of which nodes to connect and also the induced price for a link crucially depend on the distance between the involved nodes, and such settings can be modeled via edge-weighted graphs. We incorporate arbitrary edge weights by generalizing the well-known model by Fabrikant et al. [Proceedings of PODC ’03, ACM, 2003, pp. 347–351] to edge-weighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the state of the art for the unit-weight version, where the price of anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight nonconstant bound on the price of anarchy for the metric version and a slightly weaker upper bound for the nonmetric case. Moreover, we analyze the existence of equilibria, the computational hardness, and the game dynamics for several natural metrics. The model we propose can be seen as the game-theoretic analogue of the classical network design problem. Thus, low-cost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.
{"title":"Geometric Network Creation Games","authors":"Davide Bilò, Tobias Friedrich, Pascal Lenzner, Anna Melnichenko","doi":"10.1137/20m1376662","DOIUrl":"https://doi.org/10.1137/20m1376662","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 277-315, March 2024. <br/> Abstract. Network creation games are a well-known approach for explaining and analyzing the structure, quality, and dynamics of real-world networks that evolved via the interaction of selfish agents without a central authority. In these games selfish agents corresponding to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games only considered the creation of networks with unit-weight edges. In practice, e.g., when constructing a fiber-optic network, the choice of which nodes to connect and also the induced price for a link crucially depend on the distance between the involved nodes, and such settings can be modeled via edge-weighted graphs. We incorporate arbitrary edge weights by generalizing the well-known model by Fabrikant et al. [Proceedings of PODC ’03, ACM, 2003, pp. 347–351] to edge-weighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the state of the art for the unit-weight version, where the price of anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight nonconstant bound on the price of anarchy for the metric version and a slightly weaker upper bound for the nonmetric case. Moreover, we analyze the existence of equilibria, the computational hardness, and the game dynamics for several natural metrics. The model we propose can be seen as the game-theoretic analogue of the classical network design problem. Thus, low-cost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139461631","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}
Bogdan Alecu, Vadim V. Lozin, Daniel A. Quiroz, Roman Rabinovich, Igor Razgon, Viktor Zamaraev
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 261-276, March 2024. Abstract. Given two [math]-vertex graphs [math] and [math] of bounded treewidth, is there an [math]-vertex graph [math] of bounded treewidth having subgraphs isomorphic to [math] and [math]? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if [math] is a binary tree and [math] is a ternary tree. We also provide an extensive study of cases where such “gluing” is possible. In particular, we prove that if [math] has treewidth [math] and [math] has pathwidth [math], then there is an [math]-vertex graph of treewidth at most [math] containing both [math] and [math] as subgraphs.
{"title":"The Treewidth and Pathwidth of Graph Unions","authors":"Bogdan Alecu, Vadim V. Lozin, Daniel A. Quiroz, Roman Rabinovich, Igor Razgon, Viktor Zamaraev","doi":"10.1137/22m1524047","DOIUrl":"https://doi.org/10.1137/22m1524047","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 261-276, March 2024. <br/> Abstract. Given two [math]-vertex graphs [math] and [math] of bounded treewidth, is there an [math]-vertex graph [math] of bounded treewidth having subgraphs isomorphic to [math] and [math]? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if [math] is a binary tree and [math] is a ternary tree. We also provide an extensive study of cases where such “gluing” is possible. In particular, we prove that if [math] has treewidth [math] and [math] has pathwidth [math], then there is an [math]-vertex graph of treewidth at most [math] containing both [math] and [math] as subgraphs.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139411143","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 225-242, March 2024. Abstract. In this paper we prove several results on Ramsey numbers [math] for a fixed graph [math] and a large graph [math], in particular for [math]. These results extend earlier work of Erdős, Faudree, Rousseau, and Schelp and of Balister, Schelp, and Simonovits on so-called Ramsey size-linear graphs. Among other results, we show that if [math] is a subdivision of [math] with at least six vertices, then [math] for every graph [math]. We also conjecture that if [math] is a connected graph with [math], then [math]. The case [math] was proved by Erdős, Faudree, Rousseau, and Schelp. We prove the case [math].
{"title":"On Ramsey Size-Linear Graphs and Related Questions","authors":"Domagoj Bradač, Lior Gishboliner, Benny Sudakov","doi":"10.1137/22m1481713","DOIUrl":"https://doi.org/10.1137/22m1481713","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 225-242, March 2024. <br/> Abstract. In this paper we prove several results on Ramsey numbers [math] for a fixed graph [math] and a large graph [math], in particular for [math]. These results extend earlier work of Erdős, Faudree, Rousseau, and Schelp and of Balister, Schelp, and Simonovits on so-called Ramsey size-linear graphs. Among other results, we show that if [math] is a subdivision of [math] with at least six vertices, then [math] for every graph [math]. We also conjecture that if [math] is a connected graph with [math], then [math]. The case [math] was proved by Erdős, Faudree, Rousseau, and Schelp. We prove the case [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139410780","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 243-260, March 2024. Abstract. For integers [math] and [math], let [math] be the least integer [math] such that every graph with chromatic number at least [math] contains a [math]-connected subgraph with chromatic number at least [math]. Refining the recent result of Girão and Narayanan [Bull. Lond. Math. Soc., 54 (2022), pp. 868–875] that [math] for all [math], we prove that [math] for all [math] and [math]. This sharpens earlier results of Alon et al. [J. Graph Theory, 11 (1987), pp. 367–371], of Chudnovsky [J. Combin. Theory Ser. B, 103 (2013), pp. 567–586], and of Penev, Thomassé, and Trotignon [SIAM J. Discrete Math., 30 (2016), pp. 592–619]. Our result implies that [math] for all [math], making a step closer towards a conjecture of Thomassen [J. Graph Theory, 7 (1983), pp. 261–271] that [math], which was originally a result with a false proof and was the starting point of this research area.
{"title":"Highly Connected Subgraphs with Large Chromatic Number","authors":"Tung H. Nguyen","doi":"10.1137/22m150040x","DOIUrl":"https://doi.org/10.1137/22m150040x","url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 243-260, March 2024. <br/> Abstract. For integers [math] and [math], let [math] be the least integer [math] such that every graph with chromatic number at least [math] contains a [math]-connected subgraph with chromatic number at least [math]. Refining the recent result of Girão and Narayanan [Bull. Lond. Math. Soc., 54 (2022), pp. 868–875] that [math] for all [math], we prove that [math] for all [math] and [math]. This sharpens earlier results of Alon et al. [J. Graph Theory, 11 (1987), pp. 367–371], of Chudnovsky [J. Combin. Theory Ser. B, 103 (2013), pp. 567–586], and of Penev, Thomassé, and Trotignon [SIAM J. Discrete Math., 30 (2016), pp. 592–619]. Our result implies that [math] for all [math], making a step closer towards a conjecture of Thomassen [J. Graph Theory, 7 (1983), pp. 261–271] that [math], which was originally a result with a false proof and was the starting point of this research area.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139410788","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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