We prove that, with high probability, any 2-edge-colouring of a random tournament on n vertices contains a monochromatic path of length Ω(n/ √ logn). This resolves a conjecture of Ben-Eliezer, Krivelevich and Sudakov and implies a nearly tight upper bound on the oriented size Ramsey number of a directed path.
{"title":"Monochromatic paths in random tournaments","authors":"Matija Bucić, Shoham Letzter, B. Sudakov","doi":"10.1002/rsa.20780","DOIUrl":"https://doi.org/10.1002/rsa.20780","url":null,"abstract":"We prove that, with high probability, any 2-edge-colouring of a random tournament on n vertices contains a monochromatic path of length Ω(n/ √ logn). This resolves a conjecture of Ben-Eliezer, Krivelevich and Sudakov and implies a nearly tight upper bound on the oriented size Ramsey number of a directed path.","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121202705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A natural generalization of the widely discussed independent (or “internally stable”) subsets of graphs is to consider subsets of vertices where no two elements have distance less or equal to a fixed number k (“k-independent subsets”). In this paper we give asymptotic results on the average number of ˆ-independent subsets for trees of size n, where the trees are taken from a so-called simply generated family. This covers a lot of interesting examples like binary trees, general planted plane trees, and others.
{"title":"On Generalized Independent Subsets of Trees","authors":"M. Drmota, P. Kirschenhofer","doi":"10.1002/rsa.3240020204","DOIUrl":"https://doi.org/10.1002/rsa.3240020204","url":null,"abstract":"A natural generalization of the widely discussed independent (or “internally stable”) subsets of graphs is to consider subsets of vertices where no two elements have distance less or equal to a fixed number k (“k-independent subsets”). In this paper we give asymptotic results on the average number of ˆ-independent subsets for trees of size n, where the trees are taken from a so-called simply generated family. This covers a lot of interesting examples like binary trees, general planted plane trees, and others.","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"165 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120939449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The “best” inequalities of type P{(ζ, η)⊂ E} ≧f(P{η⊂ D1}P{η⊂Dm}) for independent and identically distributed random elements ζ and η can be reduced to Turan-type problems for graphs with colored vertices. In the present work we describe a finite algorithm for obtaining the asymptotical solution for an arbitrary problem of such type. In the case of two colors we obtain the final form of asymptotic solution without using the algorithm.
{"title":"Inequalities in Probability Theory and Turán-Type Problems for Graphs with Colored Vertices","authors":"A. Sidorenko","doi":"10.1002/rsa.3240020107","DOIUrl":"https://doi.org/10.1002/rsa.3240020107","url":null,"abstract":"The “best” inequalities of type P{(ζ, η)⊂ E} ≧f(P{η⊂ D1}P{η⊂Dm}) for independent and identically distributed random elements ζ and η can be reduced to Turan-type problems for graphs with colored vertices. In the present work we describe a finite algorithm for obtaining the asymptotical solution for an arbitrary problem of such type. In the case of two colors we obtain the final form of asymptotic solution without using the algorithm.","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132572353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2001-03-01DOI: 10.1002/1098-2418(200103)18:2%3C164::AID-RSA1004%3E3.0.CO;2-H
B. Pittel, R. Tungol
{"title":"A phase transition phenomenon in a random directed acyclic graph","authors":"B. Pittel, R. Tungol","doi":"10.1002/1098-2418(200103)18:2%3C164::AID-RSA1004%3E3.0.CO;2-H","DOIUrl":"https://doi.org/10.1002/1098-2418(200103)18:2%3C164::AID-RSA1004%3E3.0.CO;2-H","url":null,"abstract":"","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124729672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2001-03-01DOI: 10.1002/1098-2418(200103)18:2%3C141::AID-RSA1002%3E3.0.CO;2-W
M. Bednarska, T. Luczak
{"title":"Biased positional games and the phase transition","authors":"M. Bednarska, T. Luczak","doi":"10.1002/1098-2418(200103)18:2%3C141::AID-RSA1002%3E3.0.CO;2-W","DOIUrl":"https://doi.org/10.1002/1098-2418(200103)18:2%3C141::AID-RSA1002%3E3.0.CO;2-W","url":null,"abstract":"","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"240 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131926103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2001-01-02DOI: 10.1002/1098-2418(200101)18:1%3C39::AID-RSA4%3E3.0.CO;2-B
Dudley Stark
Poisson approximation, random graphs, Stein's method Poisson approximations for the counts of a given subgraph in large random graphs were accomplished using Stein's method by Barbour and others. Compound Poisson approximation results, on the other hand, have not appeared, at least partly because of the lack of a suitable coupling. We address that problem by introducing the concept of cluster determining pairs, leading to a useful coupling for a large class of subgraphs we call local. We find bounds on the compound Poisson approximation of counts of local subgraphs in large random graphs.
{"title":"Compound Poisson approximations of subgraph counts in random graphs","authors":"Dudley Stark","doi":"10.1002/1098-2418(200101)18:1%3C39::AID-RSA4%3E3.0.CO;2-B","DOIUrl":"https://doi.org/10.1002/1098-2418(200101)18:1%3C39::AID-RSA4%3E3.0.CO;2-B","url":null,"abstract":"Poisson approximation, random graphs, Stein's method Poisson approximations for the counts of a given subgraph in large random graphs were accomplished using Stein's method by Barbour and others. Compound Poisson approximation results, on the other hand, have not appeared, at least partly because of the lack of a suitable coupling. We address that problem by introducing the concept of cluster determining pairs, leading to a useful coupling for a large class of subgraphs we call local. We find bounds on the compound Poisson approximation of counts of local subgraphs in large random graphs.","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114275857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2001-01-02DOI: 10.1002/1098-2418(200101)18:1%3C31::AID-RSA3%3E3.0.CO;2-1
M. Luczak, C. McDiarmid
Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of ‘‘cross edges’’ between the parts. We are interested in sparse random graphs Ž . G with edge probability c n. We show that, if c ln 4, then the bisection width is n n, c n with high probability; while if c ln 4, then it is equal to 0 with high probability. There are corresponding threshold results for partitioning into any fixed number of parts. 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 31 38, 2001
考虑将图G的顶点集划分为两个大小相差不超过1的集合:G的平分宽度是所有这些部分之间“交叉边”数量划分的最小值。我们对稀疏随机图很感兴趣Ž。我们证明，如果c ln 4，则等分宽度为n n, c n具有高概率;如果c ln 4，那么它大概率等于0。对于划分为任意固定数量的部分，有相应的阈值结果。2001 John Wiley & Sons, Inc。随机结构。Alg。， 18, 31, 38, 2001
{"title":"Bisecting sparse random graphs","authors":"M. Luczak, C. McDiarmid","doi":"10.1002/1098-2418(200101)18:1%3C31::AID-RSA3%3E3.0.CO;2-1","DOIUrl":"https://doi.org/10.1002/1098-2418(200101)18:1%3C31::AID-RSA3%3E3.0.CO;2-1","url":null,"abstract":"Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of ‘‘cross edges’’ between the parts. We are interested in sparse random graphs Ž . G with edge probability c n. We show that, if c ln 4, then the bisection width is n n, c n with high probability; while if c ln 4, then it is equal to 0 with high probability. There are corresponding threshold results for partitioning into any fixed number of parts. 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 31 38, 2001","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129752574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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