{"title":"计算Gn,1/2 $$ {G}_{n,1/2} $$在度同余条件下的分区","authors":"P. Balister, Emil Powierski, A. Scott, Jane Tan","doi":"10.1002/rsa.21115","DOIUrl":null,"url":null,"abstract":"For G=Gn,1/2$$ G={G}_{n,1/2} $$ , the Erdős–Renyi random graph, let Xn$$ {X}_n $$ be the random variable representing the number of distinct partitions of V(G)$$ V(G) $$ into sets A1,…,Aq$$ {A}_1,\\dots, {A}_q $$ so that the degree of each vertex in G[Ai]$$ G\\left[{A}_i\\right] $$ is divisible by q$$ q $$ for all i∈[q]$$ i\\in \\left[q\\right] $$ . We prove that if q≥3$$ q\\ge 3 $$ is odd then Xn→dPo(1/q!)$$ {X}_n\\overset{d}{\\to \\limits}\\mathrm{Po}\\left(1/q!\\right) $$ , and if q≥4$$ q\\ge 4 $$ is even then Xn→dPo(2q/q!)$$ {X}_n\\overset{d}{\\to \\limits}\\mathrm{Po}\\left({2}^q/q!\\right) $$ . More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in G[Ai]$$ G\\left[{A}_i\\right] $$ to be congruent to xi$$ {x}_i $$ modulo q$$ q $$ for each i∈[q]$$ i\\in \\left[q\\right] $$ , where the residues xi$$ {x}_i $$ may be chosen freely. For q=2$$ q=2 $$ , the distribution is not asymptotically Poisson, but it can be determined explicitly.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"7 1","pages":"564 - 584"},"PeriodicalIF":0.9000,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Counting partitions of Gn,1/2$$ {G}_{n,1/2} $$ with degree congruence conditions\",\"authors\":\"P. Balister, Emil Powierski, A. Scott, Jane Tan\",\"doi\":\"10.1002/rsa.21115\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For G=Gn,1/2$$ G={G}_{n,1/2} $$ , the Erdős–Renyi random graph, let Xn$$ {X}_n $$ be the random variable representing the number of distinct partitions of V(G)$$ V(G) $$ into sets A1,…,Aq$$ {A}_1,\\\\dots, {A}_q $$ so that the degree of each vertex in G[Ai]$$ G\\\\left[{A}_i\\\\right] $$ is divisible by q$$ q $$ for all i∈[q]$$ i\\\\in \\\\left[q\\\\right] $$ . We prove that if q≥3$$ q\\\\ge 3 $$ is odd then Xn→dPo(1/q!)$$ {X}_n\\\\overset{d}{\\\\to \\\\limits}\\\\mathrm{Po}\\\\left(1/q!\\\\right) $$ , and if q≥4$$ q\\\\ge 4 $$ is even then Xn→dPo(2q/q!)$$ {X}_n\\\\overset{d}{\\\\to \\\\limits}\\\\mathrm{Po}\\\\left({2}^q/q!\\\\right) $$ . More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in G[Ai]$$ G\\\\left[{A}_i\\\\right] $$ to be congruent to xi$$ {x}_i $$ modulo q$$ q $$ for each i∈[q]$$ i\\\\in \\\\left[q\\\\right] $$ , where the residues xi$$ {x}_i $$ may be chosen freely. For q=2$$ q=2 $$ , the distribution is not asymptotically Poisson, but it can be determined explicitly.\",\"PeriodicalId\":54523,\"journal\":{\"name\":\"Random Structures & Algorithms\",\"volume\":\"7 1\",\"pages\":\"564 - 584\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2021-05-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Random Structures & Algorithms\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1002/rsa.21115\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, SOFTWARE ENGINEERING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures & Algorithms","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/rsa.21115","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
Counting partitions of Gn,1/2$$ {G}_{n,1/2} $$ with degree congruence conditions
For G=Gn,1/2$$ G={G}_{n,1/2} $$ , the Erdős–Renyi random graph, let Xn$$ {X}_n $$ be the random variable representing the number of distinct partitions of V(G)$$ V(G) $$ into sets A1,…,Aq$$ {A}_1,\dots, {A}_q $$ so that the degree of each vertex in G[Ai]$$ G\left[{A}_i\right] $$ is divisible by q$$ q $$ for all i∈[q]$$ i\in \left[q\right] $$ . We prove that if q≥3$$ q\ge 3 $$ is odd then Xn→dPo(1/q!)$$ {X}_n\overset{d}{\to \limits}\mathrm{Po}\left(1/q!\right) $$ , and if q≥4$$ q\ge 4 $$ is even then Xn→dPo(2q/q!)$$ {X}_n\overset{d}{\to \limits}\mathrm{Po}\left({2}^q/q!\right) $$ . More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in G[Ai]$$ G\left[{A}_i\right] $$ to be congruent to xi$$ {x}_i $$ modulo q$$ q $$ for each i∈[q]$$ i\in \left[q\right] $$ , where the residues xi$$ {x}_i $$ may be chosen freely. For q=2$$ q=2 $$ , the distribution is not asymptotically Poisson, but it can be determined explicitly.
期刊介绍:
It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness.
Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.