Pub Date : 2018-08-15DOI: 10.1017/S0963548320000267
Frank Mousset, R. Nenadov, W. Samotij
Abstract For fixed graphs F 1,…,F r , we prove an upper bound on the threshold function for the property that G(n, p) → (F 1,…,F r ). This establishes the 1-statement of a conjecture of Kohayakawa and Kreuter.
{"title":"Towards the Kohayakawa–Kreuter conjecture on asymmetric Ramsey properties","authors":"Frank Mousset, R. Nenadov, W. Samotij","doi":"10.1017/S0963548320000267","DOIUrl":"https://doi.org/10.1017/S0963548320000267","url":null,"abstract":"Abstract For fixed graphs F 1,…,F r , we prove an upper bound on the threshold function for the property that G(n, p) → (F 1,…,F r ). This establishes the 1-statement of a conjecture of Kohayakawa and Kreuter.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"54 1","pages":"943 - 955"},"PeriodicalIF":0.0,"publicationDate":"2018-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80925300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-14DOI: 10.1017/S0963548318000408
P. V. Poblete, Alfredo Viola
Thirty years ago, the Robin Hood collision resolution strategy was introduced for open addressing hash tables, and a recurrence equation was found for the distribution of its search cost. Although this recurrence could not be solved analytically, it allowed for numerical computations that, remarkably, suggested that the variance of the search cost approached a value of 1.883 when the table was full. Furthermore, by using a non-standard mean-centred search algorithm, this would imply that searches could be performed in expected constant time even in a full table. In spite of the time elapsed since these observations were made, no progress has been made in proving them. In this paper we introduce a technique to work around the intractability of the recurrence equation by solving instead an associated differential equation. While this does not provide an exact solution, it is sufficiently powerful to prove a bound of π2/3 for the variance, and thus obtain a proof that the variance of Robin Hood is bounded by a small constant for load factors arbitrarily close to 1. As a corollary, this proves that the mean-centred search algorithm runs in expected constant time. We also use this technique to study the performance of Robin Hood hash tables under a long sequence of insertions and deletions, where deletions are implemented by marking elements as deleted. We prove that, in this case, the variance is bounded by 1/(1−α), where α is the load factor. To model the behaviour of these hash tables, we use a unified approach that we apply also to study the First-Come-First-Served and Last-Come-First-Served collision resolution disciplines, both with and without deletions.
{"title":"Analysis of Robin Hood and Other Hashing Algorithms Under the Random Probing Model, With and Without Deletions","authors":"P. V. Poblete, Alfredo Viola","doi":"10.1017/S0963548318000408","DOIUrl":"https://doi.org/10.1017/S0963548318000408","url":null,"abstract":"Thirty years ago, the Robin Hood collision resolution strategy was introduced for open addressing hash tables, and a recurrence equation was found for the distribution of its search cost. Although this recurrence could not be solved analytically, it allowed for numerical computations that, remarkably, suggested that the variance of the search cost approached a value of 1.883 when the table was full. Furthermore, by using a non-standard mean-centred search algorithm, this would imply that searches could be performed in expected constant time even in a full table. In spite of the time elapsed since these observations were made, no progress has been made in proving them. In this paper we introduce a technique to work around the intractability of the recurrence equation by solving instead an associated differential equation. While this does not provide an exact solution, it is sufficiently powerful to prove a bound of π2/3 for the variance, and thus obtain a proof that the variance of Robin Hood is bounded by a small constant for load factors arbitrarily close to 1. As a corollary, this proves that the mean-centred search algorithm runs in expected constant time. We also use this technique to study the performance of Robin Hood hash tables under a long sequence of insertions and deletions, where deletions are implemented by marking elements as deleted. We prove that, in this case, the variance is bounded by 1/(1−α), where α is the load factor. To model the behaviour of these hash tables, we use a unified approach that we apply also to study the First-Come-First-Served and Last-Come-First-Served collision resolution disciplines, both with and without deletions.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"17 1","pages":"600 - 617"},"PeriodicalIF":0.0,"publicationDate":"2018-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84417312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-08DOI: 10.1017/S0963548318000573
Gwendal Collet, M. Drmota, Lukas Daniel Klausner
Abstract We prove a generalmulti-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.
{"title":"Limit laws of planar maps with prescribed vertex degrees","authors":"Gwendal Collet, M. Drmota, Lukas Daniel Klausner","doi":"10.1017/S0963548318000573","DOIUrl":"https://doi.org/10.1017/S0963548318000573","url":null,"abstract":"Abstract We prove a generalmulti-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"3 1","pages":"519 - 541"},"PeriodicalIF":0.0,"publicationDate":"2018-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87793523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-01DOI: 10.1017/S0963548318000445
Nabil H. Mustafa, Saurabh Ray
Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contain a subset of size cq + ⌊d/2⌋ whose convex hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time. In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p, q) numbers for balls in ℝd. For example, it follows from our theorem that when p > q = (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O(2d). Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.
{"title":"On a Problem of Danzer","authors":"Nabil H. Mustafa, Saurabh Ray","doi":"10.1017/S0963548318000445","DOIUrl":"https://doi.org/10.1017/S0963548318000445","url":null,"abstract":"Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contain a subset of size cq + ⌊d/2⌋ whose convex hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time. In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p, q) numbers for balls in ℝd. For example, it follows from our theorem that when p > q = (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O(2d). Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"13 1","pages":"473 - 482"},"PeriodicalIF":0.0,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81727788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-17DOI: 10.1017/S0963548320000401
R. Lang, A. Lo
Abstract Erdős, Gyárfás and Pyber showed that every r-edge-coloured complete graph Kn can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = Omega(n^{-1/(2r)})$ , then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.
{"title":"Monochromatic cycle partitions in random graphs","authors":"R. Lang, A. Lo","doi":"10.1017/S0963548320000401","DOIUrl":"https://doi.org/10.1017/S0963548320000401","url":null,"abstract":"Abstract Erdős, Gyárfás and Pyber showed that every r-edge-coloured complete graph Kn can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if \u0000$p = p(n) = Omega(n^{-1/(2r)})$\u0000 , then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"50 1","pages":"136 - 152"},"PeriodicalIF":0.0,"publicationDate":"2018-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88241216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-17DOI: 10.1017/S096354831900035X
Joshua Zahl
Abstract We prove that n plane algebraic curves determine O(n(k+2)/(k+1)) points of kth order tangency. This generalizes an earlier result of Ellenberg, Solymosi and Zahl on the number of (first order) tangencies determined by n plane algebraic curves.
{"title":"Counting higher order tangencies for plane curves","authors":"Joshua Zahl","doi":"10.1017/S096354831900035X","DOIUrl":"https://doi.org/10.1017/S096354831900035X","url":null,"abstract":"Abstract We prove that n plane algebraic curves determine O(n(k+2)/(k+1)) points of kth order tangency. This generalizes an earlier result of Ellenberg, Solymosi and Zahl on the number of (first order) tangencies determined by n plane algebraic curves.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"10 1","pages":"310 - 317"},"PeriodicalIF":0.0,"publicationDate":"2018-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79498319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-06-20DOI: 10.1017/S0963548319000403
James B. Martin, Roman Stasi'nski
Abstract We consider the behaviour of minimax recursions defined on random trees. Such recursions give the value of a general class of two-player combinatorial games. We examine in particular the case where the tree is given by a Galton–Watson branching process, truncated at some depth 2n, and the terminal values of the level 2n nodes are drawn independently from some common distribution. The case of a regular tree was previously considered by Pearl, who showed that as n → ∞ the value of the game converges to a constant, and by Ali Khan, Devroye and Neininger, who obtained a distributional limit under a suitable rescaling. For a general offspring distribution, there is a surprisingly rich variety of behaviour: the (unrescaled) value of the game may converge to a constant, or to a discrete limit with several atoms, or to a continuous distribution. We also give distributional limits under suitable rescalings in various cases. We also address questions of endogeny. Suppose the game is played on a tree with many levels, so that the terminal values are far from the root. To be confident of playing a good first move, do we need to see the whole tree and its terminal values, or can we play close to optimally by inspecting just the first few levels of the tree? The answers again depend in an interesting way on the offspring distribution. We also mention several open questions.
摘要研究了随机树上定义的极大极小递归的行为。这样的递归给出了一类一般的双人组合博弈的值。我们特别研究了这样一种情况，即树是由高尔顿-沃森分支过程给出的，在深度2n处截断，并且2n级节点的终端值是独立于一些公共分布绘制的。在正则树的情况下，Pearl证明了当n→∞时，博弈的值收敛于一个常数，Ali Khan, Devroye和Neininger在适当的重新标度下得到了一个分布极限。对于一般的后代分布，存在着令人惊讶的丰富多样的行为:游戏的(未缩放的)值可能收敛到一个常数，或者收敛到几个原子的离散极限，或者收敛到一个连续分布。在各种情况下，我们也给出了适当的重标度下的分布极限。我们还讨论了内生问题。假设游戏是在有许多关卡的树上进行的，因此终端值远离根值。为了确保第一步走得好，我们是否需要看到整个树及其最终值，或者我们是否可以通过检查树的前几层来接近最优?答案又以一种有趣的方式依赖于后代的分布。我们还提到了几个悬而未决的问题。
{"title":"Minimax functions on Galton–Watson trees","authors":"James B. Martin, Roman Stasi'nski","doi":"10.1017/S0963548319000403","DOIUrl":"https://doi.org/10.1017/S0963548319000403","url":null,"abstract":"Abstract We consider the behaviour of minimax recursions defined on random trees. Such recursions give the value of a general class of two-player combinatorial games. We examine in particular the case where the tree is given by a Galton–Watson branching process, truncated at some depth 2n, and the terminal values of the level 2n nodes are drawn independently from some common distribution. The case of a regular tree was previously considered by Pearl, who showed that as n → ∞ the value of the game converges to a constant, and by Ali Khan, Devroye and Neininger, who obtained a distributional limit under a suitable rescaling. For a general offspring distribution, there is a surprisingly rich variety of behaviour: the (unrescaled) value of the game may converge to a constant, or to a discrete limit with several atoms, or to a continuous distribution. We also give distributional limits under suitable rescalings in various cases. We also address questions of endogeny. Suppose the game is played on a tree with many levels, so that the terminal values are far from the root. To be confident of playing a good first move, do we need to see the whole tree and its terminal values, or can we play close to optimally by inspecting just the first few levels of the tree? The answers again depend in an interesting way on the offspring distribution. We also mention several open questions.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"9 1","pages":"455 - 484"},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87722386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-06-20DOI: 10.1017/S0963548319000270
R. Pinsky
Abstract For $$tau in {S_3}$$, let $$mu _n^tau $$ denote the uniformly random probability measure on the set of $$tau $$-avoiding permutations in $${S_n}$$. Let $${mathbb {N}^*} = {mathbb {N}} cup { infty } $$ with an appropriate metric and denote by $$S({mathbb{N}},{mathbb{N}^*})$$ the compact metric space consisting of functions $$sigma {rm{ = }}{ {sigma _i}} _{i = 1}^infty {rm{ }}$$ from $$mathbb {N}$$ to $${mathbb {N}^ * }$$ which are injections when restricted to $${sigma ^{ - 1}}(mathbb {N})$$; that is, if $${sigma _i}{rm{ = }}{sigma _j}$$, $$i ne j$$, then $${sigma _i} = infty $$. Extending permutations $$sigma in {S_n}$$ by defining $${sigma _j} = j$$, for $$j gt n$$, we have $${S_n} subset S({mathbb{N}},{{mathbb{N}}^*})$$. For each $$tau in {S_3}$$, we study the limiting behaviour of the measures $${ mu _n^tau } _{n = 1}^infty $$ on $$S({mathbb{N}},{mathbb{N}^*})$$. We obtain partial results for the permutation $$tau = 321$$ and complete results for the other five permutations $$tau in {S_3}$$.
对于$$tau in {S_3}$$，设$$mu _n^tau $$表示$${S_n}$$中$$tau $$ -避免排列集合上的一致随机概率测度。设$${mathbb {N}^*} = {mathbb {N}} cup { infty } $$用一个合适的度规，用$$S({mathbb{N}},{mathbb{N}^*})$$表示紧致度规空间，这个空间由从$$mathbb {N}$$到$${mathbb {N}^ * }$$的函数$$sigma {rm{ = }}{ {sigma _i}} _{i = 1}^infty {rm{ }}$$组成，当限制在$${sigma ^{ - 1}}(mathbb {N})$$时，这些函数是注入;也就是说，如果$${sigma _i}{rm{ = }}{sigma _j}$$, $$i ne j$$，那么$${sigma _i} = infty $$。通过定义$${sigma _j} = j$$扩展排列$$sigma in {S_n}$$，对于$$j gt n$$，我们有$${S_n} subset S({mathbb{N}},{{mathbb{N}}^*})$$。对于每个$$tau in {S_3}$$，我们研究了措施$${ mu _n^tau } _{n = 1}^infty $$在$$S({mathbb{N}},{mathbb{N}^*})$$上的极限行为。我们得到了该排列$$tau = 321$$的部分结果和其他五个排列$$tau in {S_3}$$的完全结果。
{"title":"The Infinite limit of random permutations avoiding patterns of length three","authors":"R. Pinsky","doi":"10.1017/S0963548319000270","DOIUrl":"https://doi.org/10.1017/S0963548319000270","url":null,"abstract":"Abstract For $$tau in {S_3}$$, let $$mu _n^tau $$ denote the uniformly random probability measure on the set of $$tau $$-avoiding permutations in $${S_n}$$. Let $${mathbb {N}^*} = {mathbb {N}} cup { infty } $$ with an appropriate metric and denote by $$S({mathbb{N}},{mathbb{N}^*})$$ the compact metric space consisting of functions $$sigma {rm{ = }}{ {sigma _i}} _{i = 1}^infty {rm{ }}$$ from $$mathbb {N}$$ to $${mathbb {N}^ * }$$ which are injections when restricted to $${sigma ^{ - 1}}(mathbb {N})$$; that is, if $${sigma _i}{rm{ = }}{sigma _j}$$, $$i ne j$$, then $${sigma _i} = infty $$. Extending permutations $$sigma in {S_n}$$ by defining $${sigma _j} = j$$, for $$j gt n$$, we have $${S_n} subset S({mathbb{N}},{{mathbb{N}}^*})$$. For each $$tau in {S_3}$$, we study the limiting behaviour of the measures $${ mu _n^tau } _{n = 1}^infty $$ on $$S({mathbb{N}},{mathbb{N}^*})$$. We obtain partial results for the permutation $$tau = 321$$ and complete results for the other five permutations $$tau in {S_3}$$.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"140 1","pages":"137 - 152"},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77724675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-06-19DOI: 10.1017/S0963548319000063
Kevin Hendrey, D. Wood
Abstract An (improper) graph colouring has defect d if each monochromatic subgraph has maximum degree at most d, and has clustering c if each monochromatic component has at most c vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than (2d+2)/(d+2)k is k-choosable with defect d. This improves upon a similar result by Havet and Sereni (J. Graph Theory, 2006). For clustered choosability of graphs with maximum average degree m, no (1-ɛ)m bound on the number of colours was previously known. The above result with d=1 solves this problem. It implies that every graph with maximum average degree m is $lfloor{frac{3}{4}m+1}rfloor$-choosable with clustering 2. This extends a result of Kopreski and Yu (Discrete Math., 2017) to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree m is $lfloor{frac{7}{10}m+1}rfloor$-choosable with clustering 9, and is $lfloor{frac{2}{3}m+1}rfloor$-choosable with clustering O(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth–moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.
{"title":"Defective and clustered choosability of sparse graphs","authors":"Kevin Hendrey, D. Wood","doi":"10.1017/S0963548319000063","DOIUrl":"https://doi.org/10.1017/S0963548319000063","url":null,"abstract":"Abstract An (improper) graph colouring has defect d if each monochromatic subgraph has maximum degree at most d, and has clustering c if each monochromatic component has at most c vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than (2d+2)/(d+2)k is k-choosable with defect d. This improves upon a similar result by Havet and Sereni (J. Graph Theory, 2006). For clustered choosability of graphs with maximum average degree m, no (1-ɛ)m bound on the number of colours was previously known. The above result with d=1 solves this problem. It implies that every graph with maximum average degree m is $lfloor{frac{3}{4}m+1}rfloor$-choosable with clustering 2. This extends a result of Kopreski and Yu (Discrete Math., 2017) to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree m is $lfloor{frac{7}{10}m+1}rfloor$-choosable with clustering 9, and is $lfloor{frac{2}{3}m+1}rfloor$-choosable with clustering O(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth–moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"113 1","pages":"791 - 810"},"PeriodicalIF":0.0,"publicationDate":"2018-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79459314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-06-15DOI: 10.1017/S0963548320000176
G. Conant
Abstract We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A 3| ≤ O(|A|), or small alternation, |AA −1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.
{"title":"On finite sets of small tripling or small alternation in arbitrary groups","authors":"G. Conant","doi":"10.1017/S0963548320000176","DOIUrl":"https://doi.org/10.1017/S0963548320000176","url":null,"abstract":"Abstract We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A 3| ≤ O(|A|), or small alternation, |AA −1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"44 1","pages":"807 - 829"},"PeriodicalIF":0.0,"publicationDate":"2018-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91122511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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