{"title":"Quasipolynomial-time algorithms for Gibbs point processes","authors":"Matthew Jenssen, Marcus Michelen, M. Ravichandran","doi":"10.1017/s0963548323000251","DOIUrl":"https://doi.org/10.1017/s0963548323000251","url":null,"abstract":"\u0000 We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a stable potential. This result holds for all activities \u0000 \u0000 \u0000 \u0000$lambda$\u0000\u0000 \u0000 for which the partition function satisfies a zero-free assumption in a neighbourhood of the interval \u0000 \u0000 \u0000 \u0000$[0,lambda ]$\u0000\u0000 \u0000 . As a corollary, for all finiterange stable potentials, we obtain a quasipolynomial-time deterministic algorithm for all \u0000 \u0000 \u0000 \u0000$lambda lt 1/(e^{B + 1} hat C_phi )$\u0000\u0000 \u0000 where \u0000 \u0000 \u0000 \u0000$hat C_phi$\u0000\u0000 \u0000 is a temperedness parameter and \u0000 \u0000 \u0000 \u0000$B$\u0000\u0000 \u0000 is the stability constant of \u0000 \u0000 \u0000 \u0000$phi$\u0000\u0000 \u0000 . In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least \u0000 \u0000 \u0000 \u0000$e^2$\u0000\u0000 \u0000 and obtain a quasipolynomial-time deterministic approximation algorithm for all \u0000 \u0000 \u0000 \u0000$lambda lt e/Delta _phi$\u0000\u0000 \u0000 , where \u0000 \u0000 \u0000 \u0000$Delta _phi$\u0000\u0000 \u0000 is the potential-weighted connective constant of the potential \u0000 \u0000 \u0000 \u0000$phi$\u0000\u0000 \u0000 . Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84454817","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}
{"title":"Expected number of faces in a random embedding of any graph is at most linear","authors":"Jesse Campion Loth, B. Mohar","doi":"10.1017/s096354832300010x","DOIUrl":"https://doi.org/10.1017/s096354832300010x","url":null,"abstract":"\u0000 A random two-cell embedding of a given graph \u0000 \u0000 \u0000 \u0000$G$\u0000\u0000 \u0000 is obtained by choosing a random local rotation around every vertex. We analyse the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl [5] proved that the expected number of faces in a random embedding of an arbitrary graph of order \u0000 \u0000 \u0000 \u0000$n$\u0000\u0000 \u0000 is at most \u0000 \u0000 \u0000 \u0000$nlog (n)$\u0000\u0000 \u0000 . While there are many families of graphs whose expected number of faces is \u0000 \u0000 \u0000 \u0000$Theta (n)$\u0000\u0000 \u0000 , none are known where the expected number would be super-linear. This led the authors of [1] to conjecture that there is a linear upper bound. In this note we confirm their conjecture by proving that for any \u0000 \u0000 \u0000 \u0000$n$\u0000\u0000 \u0000 -vertex multigraph, the expected number of faces in a random two-cell embedding is at most \u0000 \u0000 \u0000 \u0000$2nlog (2mu )$\u0000\u0000 \u0000 , where \u0000 \u0000 \u0000 \u0000$mu$\u0000\u0000 \u0000 is the maximum edge-multiplicity. This bound is best possible up to a constant factor.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"113 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84207555","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}
{"title":"A smoother notion of spread hypergraphs","authors":"Sam Spiro","doi":"10.1017/s0963548323000202","DOIUrl":"https://doi.org/10.1017/s0963548323000202","url":null,"abstract":"\u0000 Alweiss, Lovett, Wu, and Zhang introduced \u0000 \u0000 \u0000 \u0000$q$\u0000\u0000 \u0000 -spread hypergraphs in their breakthrough work regarding the sunflower conjecture, and since then \u0000 \u0000 \u0000 \u0000$q$\u0000\u0000 \u0000 -spread hypergraphs have been used to give short proofs of several outstanding problems in probabilistic combinatorics. A variant of \u0000 \u0000 \u0000 \u0000$q$\u0000\u0000 \u0000 -spread hypergraphs was implicitly used by Kahn, Narayanan, and Park to determine the threshold for when a square of a Hamiltonian cycle appears in the random graph \u0000 \u0000 \u0000 \u0000$G_{n,p}$\u0000\u0000 \u0000 . In this paper, we give a common generalization of the original notion of \u0000 \u0000 \u0000 \u0000$q$\u0000\u0000 \u0000 -spread hypergraphs and the variant used by Kahn, Narayanan, and Park.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82963122","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}
{"title":"Spanning -cycles in random graphs","authors":"Alberto Espuny Díaz, Y. Person","doi":"10.1017/s0963548323000172","DOIUrl":"https://doi.org/10.1017/s0963548323000172","url":null,"abstract":"\u0000 We extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of \u0000 \u0000 \u0000 \u0000$C_4$\u0000\u0000 \u0000 that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning \u0000 \u0000 \u0000 \u0000$K_r$\u0000\u0000 \u0000 -cycles.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"70 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91508542","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}
{"title":"On the size of maximal intersecting families","authors":"D. Zakharov","doi":"10.1017/s0963548323000287","DOIUrl":"https://doi.org/10.1017/s0963548323000287","url":null,"abstract":"\u0000 We show that an \u0000 \u0000 \u0000 \u0000$n$\u0000\u0000 \u0000 -uniform maximal intersecting family has size at most \u0000 \u0000 \u0000 \u0000$e^{-n^{0.5+o(1)}}n^n$\u0000\u0000 \u0000 . This improves a recent bound by Frankl ((2019) Comb. Probab. Comput.28(5) 733–739.). The Spread Lemma of Alweiss et al. ((2020) Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing.) plays an important role in the proof.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73433501","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}
{"title":"Constructing families of cospectral regular graphs","authors":"M. Haythorpe, Alex Newcombe","doi":"10.1017/S096354832000019X","DOIUrl":"https://doi.org/10.1017/S096354832000019X","url":null,"abstract":"Abstract A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid for special cases of a property introduced by Schwenk. For the case of cubic (3-regular) graphs, computational results are given which show that the construction generates a large proportion of the cubic graphs, which are cospectral with another cubic graph.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"5 1","pages":"664 - 671"},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72604529","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}
{"title":"Pointer chasing via triangular discrimination","authors":"A. Yehudayoff","doi":"10.1017/S0963548320000085","DOIUrl":"https://doi.org/10.1017/S0963548320000085","url":null,"abstract":"Abstract We prove an essentially sharp \u0000$tilde Omega (n/k)$\u0000 lower bound on the k-round distributional complexity of the k-step pointer chasing problem under the uniform distribution, when Bob speaks first. This is an improvement over Nisan and Wigderson’s \u0000$tilde Omega (n/{k^2})$\u0000 lower bound, and essentially matches the randomized lower bound proved by Klauck. The proof is information-theoretic, and a key part of it is using asymmetric triangular discrimination instead of total variation distance; this idea may be useful elsewhere.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"28 1","pages":"485 - 494"},"PeriodicalIF":0.0,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80724557","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}
{"title":"On subgraphs of C2k-free graphs and a problem of Kühn and Osthus","authors":"Dániel Grósz, Abhishek Methuku, C. Tompkins","doi":"10.1017/S0963548319000452","DOIUrl":"https://doi.org/10.1017/S0963548319000452","url":null,"abstract":"Abstract Let c denote the largest constant such that every C6-free graph G contains a bipartite and C4-free subgraph having a fraction c of edges of G. Győri, Kensell and Tompkins showed that 3/8 ⩽ c ⩽ 2/5. We prove that c = 38. More generally, we show that for any ε > 0, and any integer k ⩾ 2, there is a C2k-free graph \u0000\u0000$G'$\u0000\u0000 which does not contain a bipartite subgraph of girth greater than 2k with more than a fraction \u0000$$Bigl(1-frac{1}{2^{2k-2}}Bigr)frac{2}{2k-1}(1+varepsilon)$$\u0000 of the edges of \u0000\u0000$G'$\u0000\u0000. There also exists a C2k-free graph \u0000\u0000$G''$\u0000\u0000 which does not contain a bipartite and C4-free subgraph with more than a fraction \u0000$$Bigl(1-frac{1}{2^{k-1}}Bigr)frac{1}{k-1}(1+varepsilon)$$\u0000 of the edges of \u0000\u0000$G''$\u0000\u0000. One of our proofs uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erdős. For any ε > 0, and any integers a, b, k ⩾ 2, there exists an a-uniform hypergraph H of girth greater than k which does not contain any b-colourable subhypergraph with more than a fraction \u0000$$Bigl(1-frac{1}{b^{a-1}}Bigr)(1+varepsilon)$$\u0000 of the hyperedges of H. We also prove further generalizations of this theorem. In addition, we give a new and very short proof of a result of Kühn and Osthus, which states that every bipartite C2k-free graph G contains a C4-free subgraph with at least a fraction 1/(k−1) of the edges of G. We also answer a question of Kühn and Osthus about C2k-free graphs obtained by pasting together C2l’s (with k >l ⩾ 3).","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"168 1","pages":"436 - 454"},"PeriodicalIF":0.0,"publicationDate":"2020-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79533973","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}
{"title":"A note on the Brown–Erdős–Sós conjecture in groups","authors":"J. Long","doi":"10.1017/S0963548319000427","DOIUrl":"https://doi.org/10.1017/S0963548319000427","url":null,"abstract":"Abstract We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some k, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with t triples spanning \u0000$ O(sqrt t )$\u0000 vertices, which is the best possible up to the implied constant. We confirm that for all t we can find a collection of t triples spanning at most t + 3 vertices, resolving the Brown–Erdős–Sós conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemerédi’s theorem and the density Hales–Jewett theorem. This result was discovered simultaneously and independently by Nenadov, Sudakov and Tyomkyn [5], and a weaker result avoiding the arithmetic machinery was obtained independently by Wong [11].","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"32 1","pages":"633 - 640"},"PeriodicalIF":0.0,"publicationDate":"2020-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77760543","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}
{"title":"Many disjoint triangles in co-triangle-free graphs","authors":"Mykhaylo Tyomkyn","doi":"10.1017/S096354832000036X","DOIUrl":"https://doi.org/10.1017/S096354832000036X","url":null,"abstract":"Abstract We prove that any n-vertex graph whose complement is triangle-free contains n2/12 – o(n2) edge-disjoint triangles. This is tight for the disjoint union of two cliques of order n/2. We also prove a corresponding stability theorem, that all large graphs attaining the above bound are close to being bipartite. Our results answer a question of Alon and Linial, and make progress on a conjecture of Erdős.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"5 1","pages":"153 - 162"},"PeriodicalIF":0.0,"publicationDate":"2020-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89302687","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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