H. Henry, Aditiya Hermawan, E. Kusuma, Raditya Rimbawan Oprasto
Informasi merupakan hal yang penting dalam kehidupan saat ini. Peran informasi tidak dapat diabaikan sama sekali di tengah perkembangan teknologi. Semua kegiatan manusia memerlukan informasi dan bisa juga dikatakan bahwa semua kegiatan kita dituntut untuk menghasilkan informasi. Dalam dunia pendidikan, komunikasi dari pihak pihak yang terkait seperti staff pendidikan, guru, siswa dan orang tua siswa tentunya sangat berpengaruh dalam menunjang kesuksesan pendidikan. Kelancaran komunikasi, penyampaian informasi dan pengolahan informasi dari pihak pihak tersebut sangat dibutuhkan, dan dengan memanfaatkan teknologi diharapkan hal tersebut bisa dilakukan dengan lebih cepat dan akurat. Dalam sebuah Sekolah Menengah Atas seperti halnya di SMA Dharma Putra, peranan teknologi informasi sangat diperlukan seiring dengan perkembangannya. Namun pemanfaatan Teknologi Informasi belum dimanfaatkan seefektif mungkin pada SMA Dharma Putra dan masih ada sistem manual yang digunakan untuk mendukung kegiatan operasional sehari-hari, baik dalam administrasi, absensi, maupun penilaian sehingga membutuhkan waktu yang cukup lama untuk melakukan kegiatan-kegiatan tersebut. Berdasarkan permasalahan yang telah di jelaskan maka akan di lakukan Perancangan Sistem Informasi Akademik Pada SMA Dharma Putra Berbasis WEB dengan harapan dapat membantu SMA Dharma Putra dalam memberikan informasi sistem informasi akademik yang cukup bahkan lebih untuk siswa, guru dan bagian administrasi akademik sekolah.
{"title":"PERANCANGAN SISTEM INFORMASI AKADEMIK PADA SMA DHARMA PUTRA BERBASIS WEB","authors":"H. Henry, Aditiya Hermawan, E. Kusuma, Raditya Rimbawan Oprasto","doi":"10.31253/algor.v2i2.549","DOIUrl":"https://doi.org/10.31253/algor.v2i2.549","url":null,"abstract":"Informasi merupakan hal yang penting dalam kehidupan saat ini. Peran informasi tidak dapat diabaikan sama sekali di tengah perkembangan teknologi. Semua kegiatan manusia memerlukan informasi dan bisa juga dikatakan bahwa semua kegiatan kita dituntut untuk menghasilkan informasi. Dalam dunia pendidikan, komunikasi dari pihak pihak yang terkait seperti staff pendidikan, guru, siswa dan orang tua siswa tentunya sangat berpengaruh dalam menunjang kesuksesan pendidikan. Kelancaran komunikasi, penyampaian informasi dan pengolahan informasi dari pihak pihak tersebut sangat dibutuhkan, dan dengan memanfaatkan teknologi diharapkan hal tersebut bisa dilakukan dengan lebih cepat dan akurat. Dalam sebuah Sekolah Menengah Atas seperti halnya di SMA Dharma Putra, peranan teknologi informasi sangat diperlukan seiring dengan perkembangannya. Namun pemanfaatan Teknologi Informasi belum dimanfaatkan seefektif mungkin pada SMA Dharma Putra dan masih ada sistem manual yang digunakan untuk mendukung kegiatan operasional sehari-hari, baik dalam administrasi, absensi, maupun penilaian sehingga membutuhkan waktu yang cukup lama untuk melakukan kegiatan-kegiatan tersebut. Berdasarkan permasalahan yang telah di jelaskan maka akan di lakukan Perancangan Sistem Informasi Akademik Pada SMA Dharma Putra Berbasis WEB dengan harapan dapat membantu SMA Dharma Putra dalam memberikan informasi sistem informasi akademik yang cukup bahkan lebih untuk siswa, guru dan bagian administrasi akademik sekolah.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"110 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79601168","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}
We show that with high probability the random graph Gn,1/2$$ {G}_{n,1/2} $$ has an induced subgraph of linear size, all of whose degrees are congruent to r(modq)$$ rkern0.3em left(operatorname{mod}kern0.3em qright) $$ for any fixed r$$ r $$ and q≥2$$ qge 2 $$ . More generally, the same is true for any fixed distribution of degrees modulo q$$ q $$ . Finally, we show that with high probability we can partition the vertices of Gn,1/2$$ {G}_{n,1/2} $$ into q+1$$ q+1 $$ parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to r(modq)$$ rkern0.3em left(operatorname{mod}kern0.3em qright) $$ . Our results resolve affirmatively a conjecture of Scott, who addressed the case q=2$$ q=2 $$ .
{"title":"On subgraphs with degrees of prescribed residues in the random graph","authors":"Asaf Ferber, Liam Hardiman, M. Krivelevich","doi":"10.1002/rsa.21137","DOIUrl":"https://doi.org/10.1002/rsa.21137","url":null,"abstract":"We show that with high probability the random graph Gn,1/2$$ {G}_{n,1/2} $$ has an induced subgraph of linear size, all of whose degrees are congruent to r(modq)$$ rkern0.3em left(operatorname{mod}kern0.3em qright) $$ for any fixed r$$ r $$ and q≥2$$ qge 2 $$ . More generally, the same is true for any fixed distribution of degrees modulo q$$ q $$ . Finally, we show that with high probability we can partition the vertices of Gn,1/2$$ {G}_{n,1/2} $$ into q+1$$ q+1 $$ parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to r(modq)$$ rkern0.3em left(operatorname{mod}kern0.3em qright) $$ . Our results resolve affirmatively a conjecture of Scott, who addressed the case q=2$$ q=2 $$ .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"35 1","pages":"192 - 214"},"PeriodicalIF":1.0,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78351343","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}
Sofya Raskhodnikova, Noga Ron-Zewi, Nithin M. Varma
We initiate the study of the role of erasures in local decoding and use our understanding to prove a separation between erasure‐resilient and tolerant property testing. We first investigate local list‐decoding in the presence of erasures. We prove an analog of a famous result of Goldreich and Levin on local list‐decodability of the Hadamard code. Specifically, we show that the Hadamard code is locally list‐decodable in the presence of a constant fraction of erasures, arbitrarily close to 1, with list sizes and query complexity better than in the Goldreich–Levin theorem. We further study approximate locally erasure list‐decodable codes and use them to construct a property that is erasure‐resiliently testable with query complexity independent of the input length, n , but requires nΩ(1) queries for tolerant testing. We also investigate the general relationship between local decoding in the presence of errors and in the presence of erasures.
{"title":"Erasures versus errors in local decoding and property testing","authors":"Sofya Raskhodnikova, Noga Ron-Zewi, Nithin M. Varma","doi":"10.1002/rsa.21031","DOIUrl":"https://doi.org/10.1002/rsa.21031","url":null,"abstract":"We initiate the study of the role of erasures in local decoding and use our understanding to prove a separation between erasure‐resilient and tolerant property testing. We first investigate local list‐decoding in the presence of erasures. We prove an analog of a famous result of Goldreich and Levin on local list‐decodability of the Hadamard code. Specifically, we show that the Hadamard code is locally list‐decodable in the presence of a constant fraction of erasures, arbitrarily close to 1, with list sizes and query complexity better than in the Goldreich–Levin theorem. We further study approximate locally erasure list‐decodable codes and use them to construct a property that is erasure‐resiliently testable with query complexity independent of the input length, n , but requires nΩ(1) queries for tolerant testing. We also investigate the general relationship between local decoding in the presence of errors and in the presence of erasures.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"17 1","pages":"640 - 670"},"PeriodicalIF":1.0,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89571752","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}
Jie Han, Patrick Morris, Guanghui Wang, Donglei Yang
For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $rin mathbb{N}$, the $r$-independence number of $G$, denoted $alpha_r(G)$ is the largest size of a $K_r$-free set of vertices in $G$. In this paper, we discuss Ramsey--Tur'an-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal $F$-tilings. For cliques, we show that for any $kgeq 3$ and $eta>0$, any graph $G$ on $n$ vertices with $delta(G)geq eta n$ and $alpha_k(G)=o(n)$ has a $K_k$-tiling covering all but $lfloortfrac{1}{eta}rfloor(k-1)$ vertices. All conditions in this result are tight; the number of vertices left uncovered can not be improved and for $etatfrac{1}{k}$, we then show that $alpha_{k-1}(G)=o(n)$ does suffice, but not $alpha_{k-2}(G)=o(n)$. These results unify and generalise previous results of Balogh-Molla-Sharifzadeh, Nenadov-Pehova and Balogh-McDowell-Molla-Mycroft on the subject. We further explore the picture when $F$ is a tree or a cycle and discuss the effect of replacing the independence number condition with $alpha^*(G)=o(n)$ (meaning that any pair of disjoint linear sized sets induce an edge between them) where one can force perfect $F$-tilings covering all the vertices. Finally we discuss the consequences of these results in the randomly perturbed setting.
{"title":"A Ramsey–Turán theory for tilings in graphs","authors":"Jie Han, Patrick Morris, Guanghui Wang, Donglei Yang","doi":"10.1002/rsa.21182","DOIUrl":"https://doi.org/10.1002/rsa.21182","url":null,"abstract":"For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $rin mathbb{N}$, the $r$-independence number of $G$, denoted $alpha_r(G)$ is the largest size of a $K_r$-free set of vertices in $G$. In this paper, we discuss Ramsey--Tur'an-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal $F$-tilings. For cliques, we show that for any $kgeq 3$ and $eta>0$, any graph $G$ on $n$ vertices with $delta(G)geq eta n$ and $alpha_k(G)=o(n)$ has a $K_k$-tiling covering all but $lfloortfrac{1}{eta}rfloor(k-1)$ vertices. All conditions in this result are tight; the number of vertices left uncovered can not be improved and for $eta<tfrac{1}{k}$, a condition of $alpha_{k-1}(G)=o(n)$ would not suffice. When $eta>tfrac{1}{k}$, we then show that $alpha_{k-1}(G)=o(n)$ does suffice, but not $alpha_{k-2}(G)=o(n)$. These results unify and generalise previous results of Balogh-Molla-Sharifzadeh, Nenadov-Pehova and Balogh-McDowell-Molla-Mycroft on the subject. We further explore the picture when $F$ is a tree or a cycle and discuss the effect of replacing the independence number condition with $alpha^*(G)=o(n)$ (meaning that any pair of disjoint linear sized sets induce an edge between them) where one can force perfect $F$-tilings covering all the vertices. Finally we discuss the consequences of these results in the randomly perturbed setting.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"28 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76885665","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}
Kevin Steven, S. Hariyanto, Rudy Arijanto, A. Wijaya
PT. Suryaplas Intitama adalah badan usaha yang bergerak di bidang plastik, perusahaan ini mengelola data penjualan dalam jumlah yang sangat besar, didalam departemen marketing data transaksi penjualan mencapai ribuan data dalam kurun waktu satu tahun. Dalam kegiatan operasionalnya, departemen marketing menggunakan aplikasi khusus yang dibuat hanya menggunakan Microsoft Excel untuk manajemen penjualan agar dapat melihat statistik penjualan. Namun, perusahaan ini belum memiliki sebuah aplikasi dashboard untuk monitoring kegiatan proses transaksi penjualannya. Oleh karena itu, dibutuhkan pembangunan Business Intelligence untuk mengelola data di perusahaan ini kemudian dibuatkan visualisasi data dalam bentuk dashboard. Metode yang digunakan dalam perancangan data warehouse adalah metode kimball nine – step methodology. Proses ETL untuk perancangan data warehouse dilakukan dengan toolsPentaho Data Integration (PDI), sedangkan visualisasi dashboard dilakukan menggunakan aplikasi Microsoft Power BI. Hasil dari penerapan aplikasi Microsoft Power BI adalah berupa dashboard visualisasi data yang menghasilkan informasi yang dibutuhkan oleh stakeholder pada departemen marketing didalam PT. Suryaplas Intitama untuk membantu dalam pengambilan keputusan.
PT. Suryaplas本质上是一个拥有塑料业务的企业，它管理着庞大的销售数据，在市场销售部门，销售交易数据在一年内增长了数千个数据。在销售活动中，市场部只使用微软Excel作为销售管理人员开发的应用程序，以查看销售数据。然而，该公司还没有一个仪表盘应用程序来监督其销售过程。因此，需要开发商业情报来管理这家公司的数据，然后在仪表盘上创建数据可视化。设计数据仓库中使用的方法是金博九a€“一步methodology方法。平台设计数据的ETL过程是用PDI工具进行的，而仪表板可视化则使用微软功率BI应用程序进行。微软Power BI应用程序的结果是数据仪表盘可视化，它提供了PT. Suryaplas services市场部门需要的信息，以帮助决策。
{"title":"PENERAPAN BUSINESS INTELLIGENCE UNTUK MENGANALISIS DATA PADA PT. SURYAPLAS INTITAMA MENGGUNAKAN MICROSOFT POWER BI","authors":"Kevin Steven, S. Hariyanto, Rudy Arijanto, A. Wijaya","doi":"10.31253/algor.v2i2.550","DOIUrl":"https://doi.org/10.31253/algor.v2i2.550","url":null,"abstract":"\u0000 \u0000 \u0000 \u0000PT. Suryaplas Intitama adalah badan usaha yang bergerak di bidang plastik, perusahaan ini mengelola data penjualan dalam jumlah yang sangat besar, didalam departemen marketing data transaksi penjualan mencapai ribuan data dalam kurun waktu satu tahun. Dalam kegiatan operasionalnya, departemen marketing menggunakan aplikasi khusus yang dibuat hanya menggunakan Microsoft Excel untuk manajemen penjualan agar dapat melihat statistik penjualan. Namun, perusahaan ini belum memiliki sebuah aplikasi dashboard untuk monitoring kegiatan proses transaksi penjualannya. Oleh karena itu, dibutuhkan pembangunan Business Intelligence untuk mengelola data di perusahaan ini kemudian dibuatkan visualisasi data dalam bentuk dashboard. Metode yang digunakan dalam perancangan data warehouse adalah metode kimball nine – step methodology. Proses ETL untuk perancangan data warehouse dilakukan dengan toolsPentaho Data Integration (PDI), sedangkan visualisasi dashboard dilakukan menggunakan aplikasi Microsoft Power BI. Hasil dari penerapan aplikasi Microsoft Power BI adalah berupa dashboard visualisasi data yang menghasilkan informasi yang dibutuhkan oleh stakeholder pada departemen marketing didalam PT. Suryaplas Intitama untuk membantu dalam pengambilan keputusan. \u0000 \u0000 \u0000 \u0000 \u0000","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"219 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77695894","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}
Gereja Bethel Indonesia Modernland atau GBI Modernland adalah sebuah gereja di Tangerang yang bergerak melayani tuhan dan terutama mengajarkan anak-anak dalam berbagai keterampilan seperti bermusik, menari, atau melakukan sesuatu yang bisa mengembangkan bakat dari setiap individu dan edukasi pendidikan disekolah sd, smp, maupun sekolah minggu. Selama ini untuk melakukan absensi masih menggunakan fingerprint namun pengolahan data yang kurang. Penelitian ini bertujuan untuk memberikan sarana yang lebih mudah untuk anak-anak melakukan absensi pada sekolah minggu yang diharapkan bisa digunakan pada sd dan smp ataupun karyawan GBI Moderland dalam melakukan absensi. Pengumpulan data dilakukan dengan cara observasi dan wawancara juga dilakukan dengan bagian Supervisor di GBI Moderland. Aplikasi absensi dalam penelitian ini dibuat menggunakan Bahasa pemograman C Sharp.Metode penelitian yang digunakan yaitu metode Principal Component Analysis (PCA) metode yang mencari sebuah model berbasis komputer yang menggambarkan sebuah wajah dengan mengambil keterangan-keterangan yang penting dari sebuah gambar. Metode ini memiliki beberapa keunggulan salah satunya yaitu dapat bekerja secara cepat dan membutuhkan jumlah memori yang kecil. Pada aplikasi absensi ini terdapat 2 bagian yaitu frontend dan backend. Frontend merupakan halaman utama saat melakukan absensi sedangkan backend halaman untuk admin mengolah dan pengecekan data. Hasil yang diharapkan dari penelitian ini yaitu agar bisa memudahkan proses absensi dan dapat mengatasi masalah yang ada di GBI Moderland.
{"title":"RANCANG BANGUN APLIKASI ABSENSI SEKOLAH MINGGU DENGAN PENGENALAN WAJAH MENGGUNAKAN PRINCIPAL COMPONENT ANALYSIS (PCA) PADA GEREJA GBI MODERNLAND","authors":"R. Safitri, A. Susanto, R. Rino, L. Kusuma","doi":"10.31253/algor.v2i2.567","DOIUrl":"https://doi.org/10.31253/algor.v2i2.567","url":null,"abstract":"\u0000 \u0000 \u0000 \u0000Gereja Bethel Indonesia Modernland atau GBI Modernland adalah sebuah gereja di Tangerang yang bergerak melayani tuhan dan terutama mengajarkan anak-anak dalam berbagai keterampilan seperti bermusik, menari, atau melakukan sesuatu yang bisa mengembangkan bakat dari setiap individu dan edukasi pendidikan disekolah sd, smp, maupun sekolah minggu. Selama ini untuk melakukan absensi masih menggunakan fingerprint namun pengolahan data yang kurang. Penelitian ini bertujuan untuk memberikan sarana yang lebih mudah untuk anak-anak melakukan absensi pada sekolah minggu yang diharapkan bisa digunakan pada sd dan smp ataupun karyawan GBI Moderland dalam melakukan absensi. Pengumpulan data dilakukan dengan cara observasi dan wawancara juga dilakukan dengan bagian Supervisor di GBI Moderland. Aplikasi absensi dalam penelitian ini dibuat menggunakan Bahasa pemograman C Sharp.Metode penelitian yang digunakan yaitu metode Principal Component Analysis (PCA) metode yang mencari sebuah model berbasis komputer yang menggambarkan sebuah wajah dengan mengambil keterangan-keterangan yang penting dari sebuah gambar. Metode ini memiliki beberapa keunggulan salah satunya yaitu dapat bekerja secara cepat dan membutuhkan jumlah memori yang kecil. Pada aplikasi absensi ini terdapat 2 bagian yaitu frontend dan backend. Frontend merupakan halaman utama saat melakukan absensi sedangkan backend halaman untuk admin mengolah dan pengecekan data. Hasil yang diharapkan dari penelitian ini yaitu agar bisa memudahkan proses absensi dan dapat mengatasi masalah yang ada di GBI Moderland. \u0000 \u0000 \u0000 \u0000 \u0000","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"36 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86662685","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}
We show that for any d=d(n) with d0(ϵ)≤d=o(n) , with high probability, the size of a largest induced cycle in the random graph G(n,d/n) is (2±ϵ)ndlogd . This settles a long‐standing open problem in random graph theory.
{"title":"The largest hole in sparse random graphs","authors":"Nemanja Draganic, Stefan Glock, M. Krivelevich","doi":"10.1002/rsa.21078","DOIUrl":"https://doi.org/10.1002/rsa.21078","url":null,"abstract":"We show that for any d=d(n) with d0(ϵ)≤d=o(n) , with high probability, the size of a largest induced cycle in the random graph G(n,d/n) is (2±ϵ)ndlogd . This settles a long‐standing open problem in random graph theory.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"1 1","pages":"666 - 677"},"PeriodicalIF":1.0,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90289034","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}
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]$$ Gleft[{A}_iright] $$ is divisible by q$$ q $$ for all i∈[q]$$ iin left[qright] $$ . We prove that if q≥3$$ qge 3 $$ is odd then Xn→dPo(1/q!)$$ {X}_noverset{d}{to limits}mathrm{Po}left(1/q!right) $$ , and if q≥4$$ qge 4 $$ is even then Xn→dPo(2q/q!)$$ {X}_noverset{d}{to limits}mathrm{Po}left({2}^q/q!right) $$ . More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in G[Ai]$$ Gleft[{A}_iright] $$ to be congruent to xi$$ {x}_i $$ modulo q$$ q $$ for each i∈[q]$$ iin left[qright] $$ , 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.
{"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":"https://doi.org/10.1002/rsa.21115","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]$$ Gleft[{A}_iright] $$ is divisible by q$$ q $$ for all i∈[q]$$ iin left[qright] $$ . We prove that if q≥3$$ qge 3 $$ is odd then Xn→dPo(1/q!)$$ {X}_noverset{d}{to limits}mathrm{Po}left(1/q!right) $$ , and if q≥4$$ qge 4 $$ is even then Xn→dPo(2q/q!)$$ {X}_noverset{d}{to limits}mathrm{Po}left({2}^q/q!right) $$ . More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in G[Ai]$$ Gleft[{A}_iright] $$ to be congruent to xi$$ {x}_i $$ modulo q$$ q $$ for each i∈[q]$$ iin left[qright] $$ , 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":1.0,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85817721","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, Jeong Han Kim, Joonkyung Lee, T. Tran
Majority dynamics on a graph G$$ G $$ is a deterministic process such that every vertex updates its ±1$$ pm 1 $$ ‐assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnell, Tamuz and Tan conjectured that, in the Erdős–Rényi random graph G(n,p)$$ Gleft(n,pright) $$ , the random initial ±1$$ pm 1 $$ ‐assignment converges to a 99%$$ 99% $$ ‐agreement with high probability whenever p=ω(1/n)$$ p=omega left(1/nright) $$ . This conjecture was first confirmed for p≥λn−1/2$$ pge lambda {n}^{-1/2} $$ for a large constant λ$$ lambda $$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for p0$$ {lambda}^{prime }>0 $$ .
图G $$ G $$上的多数动态是一个确定性过程，使得每个顶点在每一步同时根据其邻居的多数分配更新其±1 $$ pm 1 $$‐分配。Benjamini, Chan, O'Donnell, Tamuz和Tan推测，在Erdős-Rényi随机图G(n,p) $$ Gleft(n,pright) $$中，随机初始±1 $$ pm 1 $$‐分配收敛于99%$$ 99% $$ ‐agreement with high probability whenever p=ω(1/n)$$ p=omega left(1/nright) $$ . This conjecture was first confirmed for p≥λn−1/2$$ pge lambda {n}^{-1/2} $$ for a large constant λ$$ lambda $$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for p0$$ {lambda}^{prime }>0 $$ .
{"title":"Majority dynamics on sparse random graphs","authors":"Debsoumya Chakraborti, Jeong Han Kim, Joonkyung Lee, T. Tran","doi":"10.1002/rsa.21139","DOIUrl":"https://doi.org/10.1002/rsa.21139","url":null,"abstract":"Majority dynamics on a graph G$$ G $$ is a deterministic process such that every vertex updates its ±1$$ pm 1 $$ ‐assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnell, Tamuz and Tan conjectured that, in the Erdős–Rényi random graph G(n,p)$$ Gleft(n,pright) $$ , the random initial ±1$$ pm 1 $$ ‐assignment converges to a 99%$$ 99% $$ ‐agreement with high probability whenever p=ω(1/n)$$ p=omega left(1/nright) $$ . This conjecture was first confirmed for p≥λn−1/2$$ pge lambda {n}^{-1/2} $$ for a large constant λ$$ lambda $$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for p0$$ {lambda}^{prime }>0 $$ .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"23 1","pages":"171 - 191"},"PeriodicalIF":1.0,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85818979","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}
The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Prałat noted that around p≈n−1/2$$ papprox {n}^{-1/2} $$ the clique chromatic number of the random graph Gn,p$$ {G}_{n,p} $$ changes by nΩ(1)$$ {n}^{Omega (1)} $$ when we increase the edge‐probability p$$ p $$ by no(1)$$ {n}^{o(1)} $$ , but left the details of this surprising “jump” phenomenon as an open problem. We settle this problem, that is, resolve the nature of this polynomial “jump” of the clique chromatic number of the random graph Gn,p$$ {G}_{n,p} $$ around edge‐probability p≈n−1/2$$ papprox {n}^{-1/2} $$ . Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of Gn,p$$ {G}_{n,p} $$ up to logarithmic factors for any edge‐probability p$$ p $$ .
{"title":"The jump of the clique chromatic number of random graphs","authors":"Lyuben Lichev, D. Mitsche, L. Warnke","doi":"10.1002/rsa.21128","DOIUrl":"https://doi.org/10.1002/rsa.21128","url":null,"abstract":"The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Prałat noted that around p≈n−1/2$$ papprox {n}^{-1/2} $$ the clique chromatic number of the random graph Gn,p$$ {G}_{n,p} $$ changes by nΩ(1)$$ {n}^{Omega (1)} $$ when we increase the edge‐probability p$$ p $$ by no(1)$$ {n}^{o(1)} $$ , but left the details of this surprising “jump” phenomenon as an open problem. We settle this problem, that is, resolve the nature of this polynomial “jump” of the clique chromatic number of the random graph Gn,p$$ {G}_{n,p} $$ around edge‐probability p≈n−1/2$$ papprox {n}^{-1/2} $$ . Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of Gn,p$$ {G}_{n,p} $$ up to logarithmic factors for any edge‐probability p$$ p $$ .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"1 1","pages":"1016 - 1034"},"PeriodicalIF":1.0,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91278164","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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