Adam Blumenthal, Bernard Lidick'y, Yanitsa Pehova, Florian Pfender, O. Pikhurko, Jan Volec
{"title":"Sharp bounds for decomposing graphs into edges and triangles","authors":"Adam Blumenthal, Bernard Lidick'y, Yanitsa Pehova, Florian Pfender, O. Pikhurko, Jan Volec","doi":"10.1017/S0963548320000358","DOIUrl":"https://doi.org/10.1017/S0963548320000358","url":null,"abstract":"Abstract For a real constant α, let $pi _3^alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $pi _3^alpha (n)$ be the maximum of $pi _3^alpha (G)$ over all graphs G with n vertices. The extremal function $pi _3^3(n)$ was first studied by Győri and Tuza (Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that$pi _3^3(n) le (1/2 + o(1)){n^2}$. We extend their result by determining the exact value of $pi _3^alpha (n)$ and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph ${K_{lfloor n/2 rfloor,lceil n/2 rceil }}$ are the only possible extremal examples for large n.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"42 1","pages":"271 - 287"},"PeriodicalIF":0.0,"publicationDate":"2019-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75040884","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":"Pseudorandom hypergraph matchings","authors":"S. Ehard, Stefan Glock, Felix Joos","doi":"10.1017/S0963548320000280","DOIUrl":"https://doi.org/10.1017/S0963548320000280","url":null,"abstract":"Abstract A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is ‘pseudorandom’, meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"60 1","pages":"868 - 885"},"PeriodicalIF":0.0,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84444319","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":"Erratum to ‘On Percolation and the Bunkbed Conjecture’","authors":"Svante Linusson","doi":"10.1017/S0963548319000038","DOIUrl":"https://doi.org/10.1017/S0963548319000038","url":null,"abstract":"Abstract There was an incorrect argument in the proof of the main theorem in ‘On percolation and the bunkbed conjecture’, in Combin. Probab. Comput. (2011) 20 103–117 doi: 10.1017/S0963548309990666. I thus no longer claim to have a proof for the bunkbed conjecture for outerplanar graphs.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"158 1","pages":"917 - 918"},"PeriodicalIF":0.0,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84039869","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":"An approximate version of Jackson’s conjecture","authors":"Anita Liebenau, Yanitsa Pehova","doi":"10.1017/S0963548320000152","DOIUrl":"https://doi.org/10.1017/S0963548320000152","url":null,"abstract":"Abstract A diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"12 1","pages":"886 - 899"},"PeriodicalIF":0.0,"publicationDate":"2019-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87616098","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}
Vonjy Rasendrahasina, Andry Rasoanaivo, V. Ravelomanana
{"title":"Expected Maximum Block Size in Critical Random Graphs","authors":"Vonjy Rasendrahasina, Andry Rasoanaivo, V. Ravelomanana","doi":"10.1017/S0963548319000154","DOIUrl":"https://doi.org/10.1017/S0963548319000154","url":null,"abstract":"Abstract Let G(n,M) be a uniform random graph with n vertices and M edges. Let ${wp_{n,m}}$ be the maximum block size of G(n,M), that is, the maximum size of its maximal 2-connected induced subgraphs. We determine the expectation of ${wp_{n,m}}$ near the critical point M = n/2. When n − 2M ≫ n2/3, we find a constant c1 such that $$c_1 = lim_{n rightarrow infty} left({1 - frac{2M}{n}} right) ,E({wp_{n,m}}).$$ Inside the window of transition of G(n,M) with M = (n/2)(1 + λn−1/3), where λ is any real number, we find an exact analytic expression for $$c_2(lambda) = lim_{n rightarrow infty} frac{E{left({wp_{n,{{(n/2)}({1+lambda n^{-1/3}})}}}right)}}{n^{1/3}}.$$ This study relies on the symbolic method and analytic tools from generating function theory, which enable us to describe the evolution of $n^{-1/3},E{left({wp_{n,{{(n/2)}({1+lambda n^{-1/3}})}}}right)}$ as a function of λ.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"13 1","pages":"638 - 655"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87622453","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 Erdős–Ko–Rado for random hypergraphs I","authors":"Arran Hamm, J. Kahn","doi":"10.1017/S0963548319000117","DOIUrl":"https://doi.org/10.1017/S0963548319000117","url":null,"abstract":"Abstract A family of sets is intersecting if no two of its members are disjoint, and has the Erdős–Ko–Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty intersection. Denote by ${{cal H}_k}(n,p)$ the random family in which each k-subset of {1, …, n} is present with probability p, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks: \u0000begin{equation} {rm{For what }}p = p(n,k){rm{is}}{{cal H}_k}(n,p){rm{likely to be EKR}}? end{equation}\u0000 Here, for fixed c < 1/4, and $k lt sqrt {cnlog n} $ we give a precise answer to this question, characterizing those sequences p = p(n, k) for which \u0000begin{equation} {mathbb{P}}({{cal H}_k}(n,p){rm{is EKR}}{kern 1pt} ) to 1{rm{as }}n to infty . end{equation}","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"29 1","pages":"881 - 916"},"PeriodicalIF":0.0,"publicationDate":"2019-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81147251","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":"Logarithmic Sobolev inequalities in discrete product spaces","authors":"K. Marton","doi":"10.1017/S0963548319000099","DOIUrl":"https://doi.org/10.1017/S0963548319000099","url":null,"abstract":"Abstract The aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: for a fixed probability measure q on , ( is a finite set), and any probability measure on , (*) $$D(p||q){rm{le}}C cdot sumlimits_{i = 1}^n {{rm{mathbb{E}}}_p D(p_i ( cdot |Y_1 ,{rm{ }}...,{rm{ }}Y_{i - 1} ,{rm{ }}Y_{i + 1} ,...,{rm{ }}Y_n )||q_i ( cdot |Y_1 ,{rm{ }}...,{rm{ }}Y_{i - 1} ,{rm{ }}Y_{i + 1} ,{rm{ }}...,{rm{ }}Y_n )),} $$ where pi(· |y1, …, yi−1, yi+1, …, yn) and qi(· |x1, …, xi−1, xi+1, …, xn) denote the local specifications for p resp. q, that is, the conditional distributions of the ith coordinate, given the other coordinates. The constant C depends on (the local specifications of) q. The inequality (*) ismeaningful in product spaces, in both the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for q, provided uniform logarithmic Sobolev inequalities are available for qi(· |x1, …, xi−1, xi+1, …, xn), for all fixed i and fixed (x1, …, xi−1, xi+1, …, xn). Inequality (*) directly implies that the Gibbs sampler associated with q is a contraction for relative entropy. In this paper we derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"417 1","pages":"919 - 935"},"PeriodicalIF":0.0,"publicationDate":"2019-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76463318","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 near-exponential improvement of a bound of Erdős and Lovász on maximal intersecting families","authors":"P. Frankl","doi":"10.1017/S0963548319000142","DOIUrl":"https://doi.org/10.1017/S0963548319000142","url":null,"abstract":"Abstract Let m(k) denote the maximum number of edges in a non-extendable, intersecting k-graph. Erdős and Lovász proved that m(k) ≤ kk. For k ≥ 625 we prove m(k) < kk・e−k1/4/6.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"24 1","pages":"733 - 739"},"PeriodicalIF":0.0,"publicationDate":"2019-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73904156","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 quantitative Lovász criterion for Property B","authors":"Asaf Ferber, A. Shapira","doi":"10.1017/S0963548320000334","DOIUrl":"https://doi.org/10.1017/S0963548320000334","url":null,"abstract":"Abstract A well-known observation of Lovász is that if a hypergraph is not 2-colourable, then at least one pair of its edges intersect at a single vertex. In this short paper we consider the quantitative version of Lovász’s criterion. That is, we ask how many pairs of edges intersecting at a single vertex should belong to a non-2-colourable n-uniform hypergraph. Our main result is an exact answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollobás’s two families theorem with Pluhar’s randomized colouring algorithm.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"11 1","pages":"956 - 960"},"PeriodicalIF":0.0,"publicationDate":"2019-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87857191","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":"Dirac’s theorem for random regular graphs","authors":"Padraig Condon, Alberto Espuny Díaz, António Girão, Daniela Kühn, Deryk Osthus","doi":"10.1017/S0963548320000346","DOIUrl":"https://doi.org/10.1017/S0963548320000346","url":null,"abstract":"Abstract We prove a ‘resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that whenever d is sufficiently large compared to \u0000$epsilon > 0$\u0000 , a.a.s. the following holds. Let \u0000$G'$\u0000 be any subgraph of the random n-vertex d-regular graph \u0000$G_{n,d}$\u0000 with minimum degree at least \u0000$$(1/2 + epsilon )d$$\u0000 . Then \u0000$G'$\u0000 is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly the condition that d is large cannot be omitted, and secondly the minimum degree bound cannot be improved.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"56 1","pages":"17 - 36"},"PeriodicalIF":0.0,"publicationDate":"2019-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73252486","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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