Pub Date : 2022-09-15DOI: 10.31253/algor.v4i1.1498
Fernando Verdy Sunata, S. Hariyanto, Dicky Surya Dwi Putra, Hartana Wijaya
Toko Storage merupakan nama dagang yang digunakan oleh PT. Distributor Trimitra Indonesia untuk menjual berbagai macam produk NAS (Network Attached Storage). Banyaknya produk NAS yang dijual dengan harga dan spesifikasi yang berbeda-beda, terkadang membuat bingung bahkan membuat calon konsumen kesulitan dalam memilih produk NAS yang tepat. Sehingga tidak jarang dari mereka yang bertanya mengenai rekomendasi NAS kepada admin toko. Proses pemberian rekomendasi dilakukan melalui sesi tanya jawab terkait dengan kebutuhan NAS. Proses pemberian rekomendasi terkadang memakan waktu yang lama karena harus menunggu jawaban dari calon konsumen. Karena itu, dilakukan penelitian yang bertujuan untuk merancang sebuah sistem yang mampu memberikan rekomendasi seri produk NAS kepada calon konsumen dengan menerapkan metode data mining dan algoritma multinomial naïve bayes (MNB). Hasil dari penerapan metode dan algoritma yang digunakan terbukti berhasil diimplementasikan pada data yang digunakan, hal ini dibuktikan dari hasil pengujian dan evaluasi yang dilakukan menggunakan bantuan aplikasi Weka yang menghasilkan nilai akurasi sebesar 95,5556%. Hasil akhir dari penelitian ini berupa rancangan sistem rekomendasi seri produk NAS berbasis web yang dapat digunakan oleh pengguna untuk mendapatkan rekomendasi seri produk NAS secara cepat dan tepat, hanya dengan memasukan kriteria NAS yang dicari.
{"title":"Penerapan Data Mining Untuk Merekomendasikan Seri Produk NAS Kepada Calon Konsumen Toko Storage Menggunakan Algoritma Multinomial Naïve Bayes","authors":"Fernando Verdy Sunata, S. Hariyanto, Dicky Surya Dwi Putra, Hartana Wijaya","doi":"10.31253/algor.v4i1.1498","DOIUrl":"https://doi.org/10.31253/algor.v4i1.1498","url":null,"abstract":"Toko Storage merupakan nama dagang yang digunakan oleh PT. Distributor Trimitra Indonesia untuk menjual berbagai macam produk NAS (Network Attached Storage). Banyaknya produk NAS yang dijual dengan harga dan spesifikasi yang berbeda-beda, terkadang membuat bingung bahkan membuat calon konsumen kesulitan dalam memilih produk NAS yang tepat. Sehingga tidak jarang dari mereka yang bertanya mengenai rekomendasi NAS kepada admin toko. Proses pemberian rekomendasi dilakukan melalui sesi tanya jawab terkait dengan kebutuhan NAS. Proses pemberian rekomendasi terkadang memakan waktu yang lama karena harus menunggu jawaban dari calon konsumen. Karena itu, dilakukan penelitian yang bertujuan untuk merancang sebuah sistem yang mampu memberikan rekomendasi seri produk NAS kepada calon konsumen dengan menerapkan metode data mining dan algoritma multinomial naïve bayes (MNB). Hasil dari penerapan metode dan algoritma yang digunakan terbukti berhasil diimplementasikan pada data yang digunakan, hal ini dibuktikan dari hasil pengujian dan evaluasi yang dilakukan menggunakan bantuan aplikasi Weka yang menghasilkan nilai akurasi sebesar 95,5556%. Hasil akhir dari penelitian ini berupa rancangan sistem rekomendasi seri produk NAS berbasis web yang dapat digunakan oleh pengguna untuk mendapatkan rekomendasi seri produk NAS secara cepat dan tepat, hanya dengan memasukan kriteria NAS yang dicari.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"147 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77642366","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}
Let Gn$$ {G}_n $$ be a random geometric graph, and then for q,p∈[0,1)$$ q,pin left[0,1right) $$ we construct a (q,p)$$ left(q,pright) $$ ‐perturbed noisy random geometric graph Gnq,p$$ {G}_n^{q,p} $$ where each existing edge in Gn$$ {G}_n $$ is removed with probability q$$ q $$ , while and each non‐existent edge in Gn$$ {G}_n $$ is inserted with probability p$$ p $$ . We give asymptotically tight bounds on the clique number ωGnq,p$$ omega left({G}_n^{q,p}right) $$ for several regimes of parameter.
{"title":"On the clique number of noisy random geometric graphs","authors":"Matthew Kahle, Minghao Tian, Yusu Wang","doi":"10.1002/rsa.21134","DOIUrl":"https://doi.org/10.1002/rsa.21134","url":null,"abstract":"Let Gn$$ {G}_n $$ be a random geometric graph, and then for q,p∈[0,1)$$ q,pin left[0,1right) $$ we construct a (q,p)$$ left(q,pright) $$ ‐perturbed noisy random geometric graph Gnq,p$$ {G}_n^{q,p} $$ where each existing edge in Gn$$ {G}_n $$ is removed with probability q$$ q $$ , while and each non‐existent edge in Gn$$ {G}_n $$ is inserted with probability p$$ p $$ . We give asymptotically tight bounds on the clique number ωGnq,p$$ omega left({G}_n^{q,p}right) $$ for several regimes of parameter.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"61 1 1","pages":"242 - 279"},"PeriodicalIF":1.0,"publicationDate":"2022-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90136056","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 a graph G$$ G $$ , let ck(G)$$ {c}_k(G) $$ be the number of spanning trees of G$$ G $$ with maximum degree at most k$$ k $$ . For k≥3$$ kge 3 $$ , it is proved that every connected n$$ n $$ ‐vertex r$$ r $$ ‐regular graph G$$ G $$ with r≥nk+1$$ rge frac{n}{k+1} $$ satisfies ck(G)1/n≥(1−on(1))r·zk,$$ {c}_k{(G)}^{1/n}ge left(1-{o}_n(1)right)rcdotp {z}_k, $$where zk>0$$ {z}_k>0 $$ approaches 1 extremely fast (e.g., z10=0.999971$$ {z}_{10}=0.999971 $$ ). The minimum degree requirement is essentially tight as for every k≥2$$ kge 2 $$ there are connected n$$ n $$ ‐vertex r$$ r $$ ‐regular graphs G$$ G $$ with r=⌊n/(k+1)⌋−2$$ r=leftlfloor n/left(k+1right)rightrfloor -2 $$ for which ck(G)=0$$ {c}_k(G)=0 $$ . Regularity may be relaxed, replacing r$$ r $$ with the geometric mean of the degree sequence and replacing zk$$ {z}_k $$ with zk∗>0$$ {z}_k^{ast }>0 $$ that also approaches 1, as long as the maximum degree is at most n(1−(3+ok(1))lnk/k)$$ nleft(1-left(3+{o}_k(1)right)sqrt{ln k/k}right) $$ . The same holds with no restriction on the maximum degree as long as the minimum degree is at least nk(1+ok(1))$$ frac{n}{k}left(1+{o}_k(1)right) $$ .
对于图G $$ G $$，设ck(G) $$ {c}_k(G) $$为G $$ G $$最大度不超过k $$ k $$的生成树的个数。对于k≥3 $$ kge 3 $$，证明了每个连通的n $$ n $$‐顶点r $$ r $$‐正则图G $$ G $$，且r≥nk+1 $$ rge frac{n}{k+1} $$满足ck(G)1/n≥(1−on(1))r·zk, $$ {c}_k{(G)}^{1/n}ge left(1-{o}_n(1)right)rcdotp {z}_k, $$其中zk>0 $$ {z}_k>0 $$极快地逼近1(例如z10=0.999971 $$ {z}_{10}=0.999971 $$)。最小度要求本质上是严格的，因为对于每个k≥2 $$ kge 2 $$，存在连接的n个$$ n $$‐顶点r $$ r $$‐正则图G $$ G $$，其中r=⌊n/(k+1)⌋−2 $$ r=leftlfloor n/left(k+1right)rightrfloor -2 $$，其中ck(G)=0 $$ {c}_k(G)=0 $$。正则性可以放宽，用度序列的几何平均值代替r $$ r $$，用同样趋近于1的zk * >0 $$ {z}_k^{ast }>0 $$代替zk $$ {z}_k $$，只要最大度不超过n(1−(3+ok(1))lnk/k) $$ nleft(1-left(3+{o}_k(1)right)sqrt{ln k/k}right) $$。只要最小度至少为nk(1+ok(1)) $$ frac{n}{k}left(1+{o}_k(1)right) $$，则对最大度没有限制。
{"title":"The number of bounded‐degree spanning trees","authors":"R. Yuster","doi":"10.1002/rsa.21118","DOIUrl":"https://doi.org/10.1002/rsa.21118","url":null,"abstract":"For a graph G$$ G $$ , let ck(G)$$ {c}_k(G) $$ be the number of spanning trees of G$$ G $$ with maximum degree at most k$$ k $$ . For k≥3$$ kge 3 $$ , it is proved that every connected n$$ n $$ ‐vertex r$$ r $$ ‐regular graph G$$ G $$ with r≥nk+1$$ rge frac{n}{k+1} $$ satisfies ck(G)1/n≥(1−on(1))r·zk,$$ {c}_k{(G)}^{1/n}ge left(1-{o}_n(1)right)rcdotp {z}_k, $$where zk>0$$ {z}_k>0 $$ approaches 1 extremely fast (e.g., z10=0.999971$$ {z}_{10}=0.999971 $$ ). The minimum degree requirement is essentially tight as for every k≥2$$ kge 2 $$ there are connected n$$ n $$ ‐vertex r$$ r $$ ‐regular graphs G$$ G $$ with r=⌊n/(k+1)⌋−2$$ r=leftlfloor n/left(k+1right)rightrfloor -2 $$ for which ck(G)=0$$ {c}_k(G)=0 $$ . Regularity may be relaxed, replacing r$$ r $$ with the geometric mean of the degree sequence and replacing zk$$ {z}_k $$ with zk∗>0$$ {z}_k^{ast }>0 $$ that also approaches 1, as long as the maximum degree is at most n(1−(3+ok(1))lnk/k)$$ nleft(1-left(3+{o}_k(1)right)sqrt{ln k/k}right) $$ . The same holds with no restriction on the maximum degree as long as the minimum degree is at least nk(1+ok(1))$$ frac{n}{k}left(1+{o}_k(1)right) $$ .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"45 1","pages":"737 - 757"},"PeriodicalIF":1.0,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83945251","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}
F. Benevides, C. Hoppen, H. Lefmann, Knut Odermann
The Heilbronn triangle problem is a classical geometrical problem that asks for a placement of n$$ n $$ points in the unit square [0,1]2$$ {left[0,1right]}^2 $$ that maximizes the smallest area of a triangle formed by three of those points. This problem has natural generalizations. For an integer k≥3$$ kge 3 $$ and a set 𝒫 of n$$ n $$ points in [0,1]2$$ {left[0,1right]}^2 $$ , let Ak(𝒫) be the minimum area of the convex hull of k$$ k $$ points in 𝒫 . Here, instead of considering the supremum of Ak(𝒫) over all such choices of 𝒫 , we consider its average value, Δ˜k(n)$$ {tilde{Delta}}_k(n) $$ , when the n$$ n $$ points in 𝒫 are chosen independently and uniformly at random in [0,1]2$$ {left[0,1right]}^2 $$ . We prove that Δ˜k(n)=Θn−kk−2$$ {tilde{Delta}}_k(n)=Theta left({n}^{frac{-k}{k-2}}right) $$ , for every fixed k≥3$$ kge 3 $$ .
Heilbronn三角形问题是一个经典的几何问题，它要求在单位正方形[0,1]2 $$ {left[0,1right]}^2 $$中放置n个$$ n $$点，以使由三个点组成的三角形的最小面积最大化。这个问题具有自然的普遍性。对于整数k≥3 $$ kge 3 $$，在[0,1]2 $$ {left[0,1right]}^2 $$中有n个$$ n $$点的集合，设Ak(纸牌)为集合中k个$$ k $$点的凸包面积的最小值。在这里，我们不考虑Ak(纸牌)在所有这样的选择上的最大值，而是考虑它的平均值Δ ～ k(n) $$ {tilde{Delta}}_k(n) $$，当n个$$ n $$点在[0,1]2 $$ {left[0,1right]}^2 $$中被独立且均匀随机地选择时。我们证明了Δ ～ k(n)=Θn−kk−2 $$ {tilde{Delta}}_k(n)=Theta left({n}^{frac{-k}{k-2}}right) $$，对于每一个固定k≥3 $$ kge 3 $$。
{"title":"Heilbronn triangle‐type problems in the unit square [0,1]2","authors":"F. Benevides, C. Hoppen, H. Lefmann, Knut Odermann","doi":"10.1002/rsa.21109","DOIUrl":"https://doi.org/10.1002/rsa.21109","url":null,"abstract":"The Heilbronn triangle problem is a classical geometrical problem that asks for a placement of n$$ n $$ points in the unit square [0,1]2$$ {left[0,1right]}^2 $$ that maximizes the smallest area of a triangle formed by three of those points. This problem has natural generalizations. For an integer k≥3$$ kge 3 $$ and a set 𝒫 of n$$ n $$ points in [0,1]2$$ {left[0,1right]}^2 $$ , let Ak(𝒫) be the minimum area of the convex hull of k$$ k $$ points in 𝒫 . Here, instead of considering the supremum of Ak(𝒫) over all such choices of 𝒫 , we consider its average value, Δ˜k(n)$$ {tilde{Delta}}_k(n) $$ , when the n$$ n $$ points in 𝒫 are chosen independently and uniformly at random in [0,1]2$$ {left[0,1right]}^2 $$ . We prove that Δ˜k(n)=Θn−kk−2$$ {tilde{Delta}}_k(n)=Theta left({n}^{frac{-k}{k-2}}right) $$ , for every fixed k≥3$$ kge 3 $$ .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"37 1","pages":"585 - 599"},"PeriodicalIF":1.0,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79762642","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 voter model is a classical interacting particle system modelling how consensus is formed across a network. We analyze the time to consensus for the voter model when the underlying graph is a subcritical scale‐free random graph. Moreover, we generalize the model to include a “temperature” parameter controlling how the graph influences the speed of opinion change. The interplay between the temperature and the structure of the random graph leads to a very rich phase diagram, where in the different phases different parts of the underlying geometry dominate the time to consensus. Finally, we also consider a discursive voter model, where voters discuss their opinions with their neighbors. Our proofs rely on the well‐known duality to coalescing random walks and a detailed understanding of the structure of the random graphs.
{"title":"Voter models on subcritical scale‐free random graphs","authors":"J. Fernley, Marcel Ortgiese","doi":"10.1002/rsa.21107","DOIUrl":"https://doi.org/10.1002/rsa.21107","url":null,"abstract":"The voter model is a classical interacting particle system modelling how consensus is formed across a network. We analyze the time to consensus for the voter model when the underlying graph is a subcritical scale‐free random graph. Moreover, we generalize the model to include a “temperature” parameter controlling how the graph influences the speed of opinion change. The interplay between the temperature and the structure of the random graph leads to a very rich phase diagram, where in the different phases different parts of the underlying geometry dominate the time to consensus. Finally, we also consider a discursive voter model, where voters discuss their opinions with their neighbors. Our proofs rely on the well‐known duality to coalescing random walks and a detailed understanding of the structure of the random graphs.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"18 1","pages":"376 - 429"},"PeriodicalIF":1.0,"publicationDate":"2022-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85329357","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}
A conjecture of Milena Mihail and Umesh Vazirani (Proc. 24th Annu. ACM Symp. Theory Comput., ACM, Victoria, BC, 1992, pp. 26–38.) states that the edge expansion of the graph of every polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a polytope in is greater than one over some polynomial function of . This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random polytope in is at least with high probability.
{"title":"Expansion of random 0/1 polytopes","authors":"Brett Leroux, Luis Rademacher","doi":"10.1002/rsa.21184","DOIUrl":"https://doi.org/10.1002/rsa.21184","url":null,"abstract":"A conjecture of Milena Mihail and Umesh Vazirani (Proc. 24th Annu. ACM Symp. Theory Comput., ACM, Victoria, BC, 1992, pp. 26–38.) states that the edge expansion of the graph of every polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a polytope in is greater than one over some polynomial function of . This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random polytope in is at least with high probability.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"51 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90923425","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}
Let G$$ G $$ be an n$$ n $$ ‐vertex graph, where δ(G)≥δn$$ delta (G)ge delta n $$ for some δ:=δ(n)$$ delta := delta (n) $$ . A result of Bohman, Frieze and Martin from 2003 asserts that if α(G)=Oδ2n$$ alpha (G)=Oleft({delta}^2nright) $$ , then perturbing G$$ G $$ via the addition of ωlog(1/δ)δ3$$ omega left(frac{log left(1/delta right)}{delta^3}right) $$ random edges, a.a.s. yields a Hamiltonian graph. We prove several improvements and extensions of the aforementioned result. In particular, keeping the bound on α(G)$$ alpha (G) $$ as above and allowing for δ=Ω(n−1/3)$$ delta =Omega left({n}^{-1/3}right) $$ , we determine the correct order of magnitude of the number of random edges whose addition to G$$ G $$ a.a.s. yields a pancyclic graph. Moreover, we prove similar results for sparser graphs, and assuming the correctness of Chvátal's toughness conjecture, we handle graphs having larger independent sets. Finally, under milder conditions, we determine the correct order of magnitude of the number of random edges whose addition to G$$ G $$ a.a.s. yields a graph containing an almost spanning cycle.
设G $$ G $$为n $$ n $$‐顶点图，其中δ(G)≥δn $$ delta (G)ge delta n $$对于某些δ:=δ(n) $$ delta := delta (n) $$。2003年，Bohman, Frieze和Martin的结果断言，如果α(G)=Oδ2n $$ alpha (G)=Oleft({delta}^2nright) $$，那么通过添加ωlog(1/δ)δ3 $$ omega left(frac{log left(1/delta right)}{delta^3}right) $$随机边来扰动G $$ G $$, a.a.s.产生哈密顿图。我们证明了上述结果的一些改进和扩展。特别是，如上所述，保持α(G) $$ alpha (G) $$的边界并允许δ=Ω(n−1/3)$$ delta =Omega left({n}^{-1/3}right) $$，我们确定了随机边的数量的正确数量级，这些边加到G $$ G $$ a.a.s.产生一个全环图。此外，我们证明了稀疏图的类似结果，并假设Chvátal的韧性猜想的正确性，我们处理具有更大独立集的图。最后，在较温和的条件下，我们确定了随机边的数量的正确数量级，这些边加上G $$ G $$ a.a.s.会产生一个包含几乎生成循环的图。
{"title":"Cycle lengths in randomly perturbed graphs","authors":"Elad Aigner-Horev, Dan Hefetz, M. Krivelevich","doi":"10.1002/rsa.21170","DOIUrl":"https://doi.org/10.1002/rsa.21170","url":null,"abstract":"Let G$$ G $$ be an n$$ n $$ ‐vertex graph, where δ(G)≥δn$$ delta (G)ge delta n $$ for some δ:=δ(n)$$ delta := delta (n) $$ . A result of Bohman, Frieze and Martin from 2003 asserts that if α(G)=Oδ2n$$ alpha (G)=Oleft({delta}^2nright) $$ , then perturbing G$$ G $$ via the addition of ωlog(1/δ)δ3$$ omega left(frac{log left(1/delta right)}{delta^3}right) $$ random edges, a.a.s. yields a Hamiltonian graph. We prove several improvements and extensions of the aforementioned result. In particular, keeping the bound on α(G)$$ alpha (G) $$ as above and allowing for δ=Ω(n−1/3)$$ delta =Omega left({n}^{-1/3}right) $$ , we determine the correct order of magnitude of the number of random edges whose addition to G$$ G $$ a.a.s. yields a pancyclic graph. Moreover, we prove similar results for sparser graphs, and assuming the correctness of Chvátal's toughness conjecture, we handle graphs having larger independent sets. Finally, under milder conditions, we determine the correct order of magnitude of the number of random edges whose addition to G$$ G $$ a.a.s. yields a graph containing an almost spanning cycle.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"62 1","pages":"867 - 884"},"PeriodicalIF":1.0,"publicationDate":"2022-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74130827","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}
Given a graph sequence {Gn}n≥1$$ {left{{G}_nright}}_{nge 1} $$ and a simple connected subgraph H$$ H $$ , we denote by T(H,Gn)$$ Tleft(H,{G}_nright) $$ the number of monochromatic copies of H$$ H $$ in a uniformly random vertex coloring of Gn$$ {G}_n $$ with c≥2$$ cge 2 $$ colors. We prove a central limit theorem for T(H,Gn)$$ Tleft(H,{G}_nright) $$ (we denote the appropriately centered and rescaled statistic as Z(H,Gn)$$ Zleft(H,{G}_nright) $$ ) with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of H$$ H $$ which we call good joins. Good joins are closely related to the fourth moment of Z(H,Gn)$$ Zleft(H,{G}_nright) $$ , which allows us to show a fourth moment phenomenon for the central limit theorem. For c≥30$$ cge 30 $$ , we show that Z(H,Gn)$$ Zleft(H,{G}_nright) $$ converges in distribution to 𝒩(0,1) whenever its fourth moment converges to 3. We show the convergence of the fourth moment is necessary to obtain a normal limit when c≥2$$ cge 2 $$ .
{"title":"A fourth‐moment phenomenon for asymptotic normality of monochromatic subgraphs","authors":"Sayan Das, Z. Himwich, Nitya Mani","doi":"10.1002/rsa.21166","DOIUrl":"https://doi.org/10.1002/rsa.21166","url":null,"abstract":"Given a graph sequence {Gn}n≥1$$ {left{{G}_nright}}_{nge 1} $$ and a simple connected subgraph H$$ H $$ , we denote by T(H,Gn)$$ Tleft(H,{G}_nright) $$ the number of monochromatic copies of H$$ H $$ in a uniformly random vertex coloring of Gn$$ {G}_n $$ with c≥2$$ cge 2 $$ colors. We prove a central limit theorem for T(H,Gn)$$ Tleft(H,{G}_nright) $$ (we denote the appropriately centered and rescaled statistic as Z(H,Gn)$$ Zleft(H,{G}_nright) $$ ) with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of H$$ H $$ which we call good joins. Good joins are closely related to the fourth moment of Z(H,Gn)$$ Zleft(H,{G}_nright) $$ , which allows us to show a fourth moment phenomenon for the central limit theorem. For c≥30$$ cge 30 $$ , we show that Z(H,Gn)$$ Zleft(H,{G}_nright) $$ converges in distribution to 𝒩(0,1) whenever its fourth moment converges to 3. We show the convergence of the fourth moment is necessary to obtain a normal limit when c≥2$$ cge 2 $$ .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"80 1","pages":"968 - 996"},"PeriodicalIF":1.0,"publicationDate":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91143161","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}
A well‐known result of Ajtai Komlós, Pintz, Spencer, and Szemerédi (J. Combin. Theory Ser. A 32 (1982), 321–335) states that every k$$ k $$ ‐graph H$$ H $$ on n$$ n $$ vertices, with girth at least five, and average degree tk−1$$ {t}^{k-1} $$ contains an independent set of size cn(logt)1/(k−1)/t$$ cn{left(log tright)}^{1/left(k-1right)}/t $$ for some c>0$$ c>0 $$ . In this paper we show that an independent set of the same size can be found under weaker conditions allowing certain cycles of length 2, 3, and 4. Our work is motivated by a problem of Lo and Zhao, who asked for k≥4$$ kge 4 $$ , how large of an independent set a k$$ k $$ ‐graph H$$ H $$ on n$$ n $$ vertices necessarily has when its maximum (k−2)$$ left(k-2right) $$ ‐degree Δk−2(H)≤dn$$ {Delta}_{k-2}(H)le dn $$ . (The corresponding problem with respect to (k−1)$$ left(k-1right) $$ ‐degrees was solved by Kostochka, Mubayi, and Verstraëte (Random Struct. & Algorithms 44 (2014), 224–239).) In this paper we show that every k$$ k $$ ‐graph H$$ H $$ on n$$ n $$ vertices with Δk−2(H)≤dn$$ {Delta}_{k-2}(H)le dn $$ contains an independent set of size cndloglognd1/(k−1)$$ c{left(frac{n}{d}mathrm{loglog}frac{n}{d}right)}^{1/left(k-1right)} $$ , and under additional conditions, an independent set of size cndlognd1/(k−1)$$ c{left(frac{n}{d}log frac{n}{d}right)}^{1/left(k-1right)} $$ . The former assertion gives a new upper bound for the (k−2)$$ left(k-2right) $$ ‐degree Turán density of complete k$$ k $$ ‐graphs.
Ajtai Komlós、Pintz、Spencer和szemersamudi (J. Combin)的一个众所周知的结果。理论SerA 32(1982)， 321-335)指出，每个k $$ k $$‐图H $$ H $$在n个$$ n $$顶点上，周长至少为5，平均度为tk−1 $$ {t}^{k-1} $$包含一个大小为cn(logt)1/(k−1)/t $$ cn{left(log tright)}^{1/left(k-1right)}/t $$的独立集合，对于某些c>0 $$ c>0 $$。在本文中，我们证明了在允许长度为2,3和4的某些循环的较弱条件下，可以找到相同大小的独立集。我们的工作受到Lo和Zhao的一个问题的启发，他们要求k≥4 $$ kge 4 $$，当k $$ k $$‐图H $$ H $$的最大值(k−2)$$ left(k-2right) $$‐度Δk−2(H)≤dn $$ {Delta}_{k-2}(H)le dn $$时，n个$$ n $$顶点上的k ‐图H 的独立集有多大。(关于(k−1)$$ left(k-1right) $$‐degrees的相应问题由Kostochka, Mubayi和Verstraëte (Random Struct)解决。＆算法44(2014)，224-239)。在本文中，我们证明了在Δk−2(H)≤dn $$ {Delta}_{k-2}(H)le dn $$的n个$$ n $$顶点上的每k $$ k $$‐图H $$ H $$包含一个大小为cndloggnd1 /(k−1)$$ c{left(frac{n}{d}mathrm{loglog}frac{n}{d}right)}^{1/left(k-1right)} $$的独立集，并且在附加条件下，包含一个大小为cndloggnd1 /(k−1)$$ c{left(frac{n}{d}log frac{n}{d}right)}^{1/left(k-1right)} $$的独立集。前一个断言给出了完全k $$ k $$‐图的(k−2)$$ left(k-2right) $$‐度Turán密度的一个新的上界。
{"title":"Independence number of hypergraphs under degree conditions","authors":"V. Rödl, M. Sales, Yi Zhao","doi":"10.1002/rsa.21151","DOIUrl":"https://doi.org/10.1002/rsa.21151","url":null,"abstract":"A well‐known result of Ajtai Komlós, Pintz, Spencer, and Szemerédi (J. Combin. Theory Ser. A 32 (1982), 321–335) states that every k$$ k $$ ‐graph H$$ H $$ on n$$ n $$ vertices, with girth at least five, and average degree tk−1$$ {t}^{k-1} $$ contains an independent set of size cn(logt)1/(k−1)/t$$ cn{left(log tright)}^{1/left(k-1right)}/t $$ for some c>0$$ c>0 $$ . In this paper we show that an independent set of the same size can be found under weaker conditions allowing certain cycles of length 2, 3, and 4. Our work is motivated by a problem of Lo and Zhao, who asked for k≥4$$ kge 4 $$ , how large of an independent set a k$$ k $$ ‐graph H$$ H $$ on n$$ n $$ vertices necessarily has when its maximum (k−2)$$ left(k-2right) $$ ‐degree Δk−2(H)≤dn$$ {Delta}_{k-2}(H)le dn $$ . (The corresponding problem with respect to (k−1)$$ left(k-1right) $$ ‐degrees was solved by Kostochka, Mubayi, and Verstraëte (Random Struct. & Algorithms 44 (2014), 224–239).) In this paper we show that every k$$ k $$ ‐graph H$$ H $$ on n$$ n $$ vertices with Δk−2(H)≤dn$$ {Delta}_{k-2}(H)le dn $$ contains an independent set of size cndloglognd1/(k−1)$$ c{left(frac{n}{d}mathrm{loglog}frac{n}{d}right)}^{1/left(k-1right)} $$ , and under additional conditions, an independent set of size cndlognd1/(k−1)$$ c{left(frac{n}{d}log frac{n}{d}right)}^{1/left(k-1right)} $$ . The former assertion gives a new upper bound for the (k−2)$$ left(k-2right) $$ ‐degree Turán density of complete k$$ k $$ ‐graphs.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"6 1","pages":"821 - 863"},"PeriodicalIF":1.0,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83050585","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 prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree-child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (2022). For patterns of height $1$ and $2$, we show that they either occur frequently (mean is asymptotically linear and limit law is normal) or sporadically (mean is asymptotically constant and limit law is Poisson) or not all (mean tends to $0$ and limit law is degenerate). We expect that these are the only possible limit laws for any fringe pattern.
{"title":"Limit theorems for patterns in ranked tree‐child networks","authors":"Michael Fuchs, Hexuan Liu, Tsan-Cheng Yu","doi":"10.1002/rsa.21177","DOIUrl":"https://doi.org/10.1002/rsa.21177","url":null,"abstract":"We prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree-child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (2022). For patterns of height $1$ and $2$, we show that they either occur frequently (mean is asymptotically linear and limit law is normal) or sporadically (mean is asymptotically constant and limit law is Poisson) or not all (mean tends to $0$ and limit law is degenerate). We expect that these are the only possible limit laws for any fringe pattern.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83529908","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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