Pub Date : 2021-01-18DOI: 10.2174/1877946811666210118121319
J. Travadi
A statistical study between ‘Mesophase Lower Transition Temperatures’ (MLTTs) and their structural properties is carried out to understand the effect of structural behaviour on mesomorphic property. To establish a “Quantitative Structure and Property Relationship (QSPR) model” a set of randomly selected thirty-nine mesomorphic compounds is constructed. The backward stepwise regression analysis method is used to find out the good correlation between the “Mesophase Lower Transition Temperatures (MLTTs)” data set and “physical descriptors” like AMR, bpol, ASP-0, DELS, SdssC, etc. Physical descriptors are selected based on their good r2 -values and p-values with respective MLTTs. The derived QSPR equation shows a good correlation between structural properties and mesomorphic properties of compounds. Validation of the derived QSPR equation is carried out on the test series of eight compounds. The MLTTs of these compounds are predicted through the statistically derived QSPR equation and then compared with experimentally measured MLTTs. The average percentage error observed between predicted MLTTs and experimentally measured MLTTs is observed 10.95 % for all the thirty-nine compounds of the trial set, and 10.64% for 8 compounds of the test series respectively. A low average percentage error suggests a reasonably acceptable degree of accuracy of the generated QSPR model to predict MLTTs of the compounds having a similar type of structure.In the present study not only MLTTs are predicted, but an effort also made to predict “Latent Transition Temperatures” (LTTs) of some non-mesomorphic compounds from derived QSPR equation. This computational study gives a sight to develop new QSPR models for the different-different type of liquid crystals homologous series, through which various types of mesomorphic properties, like mesomorphic thermal stability, mesomorphic upper transition temperature, mesophase length, phase behaviour, etc. can study and predict.
{"title":"Statistical Study of Relationship between Structural Properties and Mesomorphic Properties of Some Ester Linkage Mesomorphic Compounds","authors":"J. Travadi","doi":"10.2174/1877946811666210118121319","DOIUrl":"https://doi.org/10.2174/1877946811666210118121319","url":null,"abstract":"\u0000\u0000A statistical study between ‘Mesophase Lower Transition Temperatures’ (MLTTs) and their structural properties\u0000is carried out to understand the effect of structural behaviour on mesomorphic property.\u0000\u0000\u0000\u0000To establish a “Quantitative Structure and Property Relationship (QSPR) model” a set of randomly selected\u0000thirty-nine mesomorphic compounds is constructed. The backward stepwise regression analysis method is used to find out\u0000the good correlation between the “Mesophase Lower Transition Temperatures (MLTTs)” data set and “physical descriptors”\u0000like AMR, bpol, ASP-0, DELS, SdssC, etc. Physical descriptors are selected based on their good r2\u0000-values and p-values\u0000with respective MLTTs. The derived QSPR equation shows a good correlation between structural properties and mesomorphic properties of compounds.\u0000\u0000\u0000\u0000 Validation of the derived QSPR equation is carried out on the test series of eight compounds. The MLTTs of these\u0000compounds are predicted through the statistically derived QSPR equation and then compared with experimentally measured\u0000MLTTs. The average percentage error observed between predicted MLTTs and experimentally measured MLTTs is observed 10.95 % for all the thirty-nine compounds of the trial set, and 10.64% for 8 compounds of the test series respectively.\u0000\u0000\u0000\u0000 A low average percentage error suggests a reasonably acceptable degree of accuracy of the generated\u0000QSPR model to predict MLTTs of the compounds having a similar type of structure.In the present study not only MLTTs\u0000are predicted, but an effort also made to predict “Latent Transition Temperatures” (LTTs) of some non-mesomorphic compounds from derived QSPR equation.\u0000\u0000\u0000\u0000This computational study gives a sight to develop new QSPR models for the different-different type of liquid\u0000crystals homologous series, through which various types of mesomorphic properties, like mesomorphic thermal stability,\u0000mesomorphic upper transition temperature, mesophase length, phase behaviour, etc. can study and predict.\u0000","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45430948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-13DOI: 10.1017/s0963548321000584
Umberto De Ambroggio, Matthew I. Roberts
We consider the near-critical Erdős–Rényi random graph G(n, p) and provide a new probabilistic proof of the fact that, when p is of the form $p=p(n)=1/n+lambda/n^{4/3}$ and A is large, begin{equation*}mathbb{P}(|mathcal{C}_{max}|>An^{2/3})asymp A^{-3/2}e^{-frac{A^3}{8}+frac{lambda A^2}{2}-frac{lambda^2A}{2}},end{equation*} where $mathcal{C}_{max}$ is the largest connected component of the graph. Our result allows A and $lambda$ to depend on n. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erdős–Rényi graphs, together with analytic estimates.
{"title":"Unusually large components in near-critical Erdős-Rényi graphs via ballot theorems","authors":"Umberto De Ambroggio, Matthew I. Roberts","doi":"10.1017/s0963548321000584","DOIUrl":"https://doi.org/10.1017/s0963548321000584","url":null,"abstract":"\u0000 We consider the near-critical Erdős–Rényi random graph G(n, p) and provide a new probabilistic proof of the fact that, when p is of the form \u0000 \u0000 \u0000 \u0000$p=p(n)=1/n+lambda/n^{4/3}$\u0000\u0000 \u0000 and A is large,\u0000\u0000 \u0000 \u0000 \u0000begin{equation*}mathbb{P}(|mathcal{C}_{max}|>An^{2/3})asymp A^{-3/2}e^{-frac{A^3}{8}+frac{lambda A^2}{2}-frac{lambda^2A}{2}},end{equation*}\u0000\u0000 \u0000 where \u0000 \u0000 \u0000 \u0000$mathcal{C}_{max}$\u0000\u0000 \u0000 is the largest connected component of the graph. Our result allows A and \u0000 \u0000 \u0000 \u0000$lambda$\u0000\u0000 \u0000 to depend on n. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erdős–Rényi graphs, together with analytic estimates.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76937181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1017/S0963548320000413
Asaf Ferber, K. Luh, Gweneth McKinley
{"title":"Triangle-degrees in graphs and tetrahedron coverings in 3-graphs","authors":"Asaf Ferber, K. Luh, Gweneth McKinley","doi":"10.1017/S0963548320000413","DOIUrl":"https://doi.org/10.1017/S0963548320000413","url":null,"abstract":"","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75788662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
С. Х. Акназаров, А. Б. Сейсенова, У.Д. Диканбай, А.Ж. Мутушев, Хуан Мария Гонсалес-Лил
В работе исследована возможность получения диборида алюминия методом самораспространяющегося высокотемпературного синтеза (СВС), рассчитаны термодинамические параметры синтеза, выбран компонентный состав шихты. Экспериментально подобран количественный состав шихты для получения в процессе синтеза диборида алюминия, отвечающего предъявляемыми к нему требованиями, с максимальным выходом.
{"title":"Получение металлического горючего AlB2 методом самораспространяющегося высокотемпературного синтеза и высокоэнергетические составы на его основе","authors":"С. Х. Акназаров, А. Б. Сейсенова, У.Д. Диканбай, А.Ж. Мутушев, Хуан Мария Гонсалес-Лил","doi":"10.18321/CPC402","DOIUrl":"https://doi.org/10.18321/CPC402","url":null,"abstract":"В работе исследована возможность получения диборида алюминия методом самораспространяющегося высокотемпературного синтеза (СВС), рассчитаны термодинамические параметры синтеза, выбран компонентный состав шихты. Экспериментально подобран количественный состав шихты для получения в процессе синтеза диборида алюминия, отвечающего предъявляемыми к нему требованиями, с максимальным выходом.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43861441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-10DOI: 10.1017/S0963548322000165
S. Norin, A. Scott, D. Wood
The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.
{"title":"Clustered colouring of graph classes with bounded treedepth or pathwidth","authors":"S. Norin, A. Scott, D. Wood","doi":"10.1017/S0963548322000165","DOIUrl":"https://doi.org/10.1017/S0963548322000165","url":null,"abstract":"The \"clustered chromatic number\" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73419589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-03DOI: 10.1017/S0963548321000092
B. Green, A. Walker
We prove that if $A subseteq [X,,2X]$ and $B subseteq [Y,,2Y]$ are sets of integers such that gcd (a, b) ⩾ D for at least δ|A||B| pairs (a, b) ε A × B then $|A||B|{ ll _{rm{varepsilon }}}{delta ^{ - 2 - varepsilon }}XY/{D^2}$ . This is a new result even when δ = 1. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.
我们证明，如果$A subseteq [X,,2X]$和$B subseteq [Y,,2Y]$是整数集，使得gcd (a, b)小于或等于D至少δ| a || b |对(a, b) ε a × b，那么$|A||B|{ ll _{rm{varepsilon }}}{delta ^{ - 2 - varepsilon }}XY/{D^2}$。即使δ = 1，这也是一个新结果。这个证明使用了Koukoulopoulos和Maynard的思想以及一些额外的组合论证。
{"title":"Extremal problems for GCDs","authors":"B. Green, A. Walker","doi":"10.1017/S0963548321000092","DOIUrl":"https://doi.org/10.1017/S0963548321000092","url":null,"abstract":"\u0000 We prove that if \u0000 \u0000 \u0000 \u0000$A subseteq [X,,2X]$\u0000\u0000 \u0000 and \u0000 \u0000 \u0000 \u0000$B subseteq [Y,,2Y]$\u0000\u0000 \u0000 are sets of integers such that gcd (a, b) ⩾ D for at least δ|A||B| pairs (a, b) ε A × B then \u0000 \u0000 \u0000 \u0000$|A||B|{ ll _{rm{varepsilon }}}{delta ^{ - 2 - varepsilon }}XY/{D^2}$\u0000\u0000 \u0000 . This is a new result even when δ = 1. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86250512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-03DOI: 10.1017/s0963548322000074
Andrzej Grzesik, Joonkyung Lee, Bernard Lidick'y, Jan Volec
A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring. Burr and Rosta, extending a famous conjecture of Erdős, conjectured that every graph is common. The conjectures of Erdős and of Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples of common graphs had not seen much progress since then, although very recently a few more graphs were verified to be common by the flag algebra method or the recent progress on Sidorenko’s conjecture. Our contribution here is to provide several new classes of tripartite common graphs. The first example is the class of so-called triangle trees, which generalises two theorems by Sidorenko and answers a question of Jagger, Šťovíček, and Thomason from 1996. We also prove that, somewhat surprisingly, given any tree $T$ , there exists a triangle tree such that the graph obtained by adding $T$ as a pendant tree is still common. Furthermore, we show that adding arbitrarily many apex vertices to any connected bipartite graph on at most $5$ vertices yields a common graph.
{"title":"On tripartite common graphs","authors":"Andrzej Grzesik, Joonkyung Lee, Bernard Lidick'y, Jan Volec","doi":"10.1017/s0963548322000074","DOIUrl":"https://doi.org/10.1017/s0963548322000074","url":null,"abstract":"\u0000 A graph \u0000 \u0000 \u0000 \u0000$H$\u0000\u0000 \u0000 is common if the number of monochromatic copies of \u0000 \u0000 \u0000 \u0000$H$\u0000\u0000 \u0000 in a 2-edge-colouring of the complete graph \u0000 \u0000 \u0000 \u0000$K_n$\u0000\u0000 \u0000 is asymptotically minimised by the random colouring. Burr and Rosta, extending a famous conjecture of Erdős, conjectured that every graph is common. The conjectures of Erdős and of Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples of common graphs had not seen much progress since then, although very recently a few more graphs were verified to be common by the flag algebra method or the recent progress on Sidorenko’s conjecture. Our contribution here is to provide several new classes of tripartite common graphs. The first example is the class of so-called triangle trees, which generalises two theorems by Sidorenko and answers a question of Jagger, Šťovíček, and Thomason from 1996. We also prove that, somewhat surprisingly, given any tree \u0000 \u0000 \u0000 \u0000$T$\u0000\u0000 \u0000 , there exists a triangle tree such that the graph obtained by adding \u0000 \u0000 \u0000 \u0000$T$\u0000\u0000 \u0000 as a pendant tree is still common. Furthermore, we show that adding arbitrarily many apex vertices to any connected bipartite graph on at most \u0000 \u0000 \u0000 \u0000$5$\u0000\u0000 \u0000 vertices yields a common graph.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83875851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-18DOI: 10.1017/S0963548321000298
Martin Kurečka
A set S of permutations is forcing if for any sequence ${Pi_i}_{i in mathbb{N}}$ of permutations where the density $d(pi,Pi_i)$ converges to $frac{1}{|pi|!}$ for every permutation $pi in S$ , it holds that ${Pi_i}_{i in mathbb{N}}$ is quasirandom. Graham asked whether there exists an integer k such that the set of all permutations of order k is forcing; this has been shown to be true for any $kge 4$ . In particular, the set of all 24 permutations of order 4 is forcing. We provide the first non-trivial lower bound on the size of a forcing set of permutations: every forcing set of permutations (with arbitrary orders) contains at least four permutations.
一组S的排列是强制的，如果对于任意排列序列${Pi_i}_{i in mathbb{N}}$，其中密度$d(pi,Pi_i)$收敛于$frac{1}{|pi|!}$对于每个排列$pi in S$，则认为${Pi_i}_{i in mathbb{N}}$是准随机的。Graham问是否存在一个整数k使得所有k阶排列的集合是强制的;这已被证明对任何$kge 4$都是正确的。特别地，所有24个4阶排列的集合是强制的。我们提供了排列强迫集大小的第一个非平凡下界:每个排列强迫集(任意顺序)至少包含四个排列。
{"title":"Lower bound on the size of a quasirandom forcing set of permutations","authors":"Martin Kurečka","doi":"10.1017/S0963548321000298","DOIUrl":"https://doi.org/10.1017/S0963548321000298","url":null,"abstract":"\u0000 A set S of permutations is forcing if for any sequence \u0000 \u0000 \u0000 \u0000${Pi_i}_{i in mathbb{N}}$\u0000\u0000 \u0000 of permutations where the density \u0000 \u0000 \u0000 \u0000$d(pi,Pi_i)$\u0000\u0000 \u0000 converges to \u0000 \u0000 \u0000 \u0000$frac{1}{|pi|!}$\u0000\u0000 \u0000 for every permutation \u0000 \u0000 \u0000 \u0000$pi in S$\u0000\u0000 \u0000 , it holds that \u0000 \u0000 \u0000 \u0000${Pi_i}_{i in mathbb{N}}$\u0000\u0000 \u0000 is quasirandom. Graham asked whether there exists an integer k such that the set of all permutations of order k is forcing; this has been shown to be true for any \u0000 \u0000 \u0000 \u0000$kge 4$\u0000\u0000 \u0000 . In particular, the set of all 24 permutations of order 4 is forcing. We provide the first non-trivial lower bound on the size of a forcing set of permutations: every forcing set of permutations (with arbitrary orders) contains at least four permutations.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89340412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-15DOI: 10.1017/S0963548322000153
Julia Böttcher, Olaf Parczyk, Amedeo Sgueglia, J. Skokan
We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$-vertex graph $G$ with linear minimum degree and the binomial random graph $G(n,p)$. We prove that asymptotically almost surely $G cup G(n,p)$ contains $min {delta(G), lfloor n/3 rfloor }$ pairwise vertex-disjoint triangles, provided $p ge C log n/n$, where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, to appear] in the case of triangle-factors. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159-176] this fully resolves the existence of triangle-factors in randomly perturbed graphs. We also prove a stability version of our result. Finally, we discuss further generalisations to larger clique-factors, larger cycle-factors, and $2$-universality.
研究了任意具有线性最小度的$n$ -顶点图$G$与二项随机图$G(n,p)$的并集的随机摄动图模型中寻找两两顶点不相交三角形的问题。我们渐近地几乎肯定地证明$G cup G(n,p)$包含$min {delta(G), lfloor n/3 rfloor }$对顶点不相交三角形，假设$p ge C log n/n$，其中$C$是一个足够大的常数。这是狄拉克旧结果的扰动版本。我们的结果是渐近最优的，并回答了Han, Morris和Treglown [RSA，出现]在三角形因子的情况下的问题。连同Balogh, treglon和Wagner的结果[CPC, 2019, no. 5]。[2,159 -176]这完全解决了随机摄动图中三角形因子的存在性。我们还证明了结果的稳定性版本。最后，我们讨论了进一步推广到更大的团因子、更大的循环因子和$2$ -普遍性。
{"title":"Triangles in randomly perturbed graphs","authors":"Julia Böttcher, Olaf Parczyk, Amedeo Sgueglia, J. Skokan","doi":"10.1017/S0963548322000153","DOIUrl":"https://doi.org/10.1017/S0963548322000153","url":null,"abstract":"We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$-vertex graph $G$ with linear minimum degree and the binomial random graph $G(n,p)$. We prove that asymptotically almost surely $G cup G(n,p)$ contains $min {delta(G), lfloor n/3 rfloor }$ pairwise vertex-disjoint triangles, provided $p ge C log n/n$, where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, to appear] in the case of triangle-factors. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159-176] this fully resolves the existence of triangle-factors in randomly perturbed graphs. We also prove a stability version of our result. Finally, we discuss further generalisations to larger clique-factors, larger cycle-factors, and $2$-universality.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87722905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-13DOI: 10.1017/S0963548322000025
Ivailo Hartarsky, Laure Marech'e
We study a general class of interacting particle systems called kinetically constrained models (KCM) in two dimensions tightly linked to the monotone cellular automata called bootstrap percolation. There are three classes of such models, the most studied being the critical one. In a recent series of works by Martinelli, Morris, Toninelli and the authors, it was shown that the KCM counterparts of critical bootstrap percolation models with the same properties split into two classes with different behaviour. Together with the companion paper by the first author, our work determines the logarithm of the infection time up to a constant factor for all critical KCM, which were previously known only up to logarithmic corrections. This improves all previous results except for the Duarte-KCM, for which we give a new proof of the best result known. We establish that on this level of precision critical KCM have to be classified into seven categories instead of the two in bootstrap percolation. In the present work, we establish lower bounds for critical KCM in a unified way, also recovering the universality result of Toninelli and the authors and the Duarte model result of Martinelli, Toninelli and the second author.
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