{"title":"A central limit theorem for additive functionals of increasing trees","authors":"D. Ralaivaosaona, S. Wagner","doi":"10.1017/S0963548318000585","DOIUrl":"https://doi.org/10.1017/S0963548318000585","url":null,"abstract":"Abstract A tree functional is called additive if it satisfies a recursion of the form $F(T) = sum_{j=1}^k F(B_j) + f(T)$, where B1, …, Bk are the branches of the tree T and f (T) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalized plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of d-ary increasing trees, but other examples (old and new) are covered as well.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"65 1","pages":"618 - 637"},"PeriodicalIF":0.0,"publicationDate":"2019-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90862817","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":"Decomposing edge-coloured graphs under colour degree constraints","authors":"S. Fujita, Ruonan Li, Guanghui Wang","doi":"10.1017/S0963548319000014","DOIUrl":"https://doi.org/10.1017/S0963548319000014","url":null,"abstract":"Abstract For an edge-coloured graph G, the minimum colour degree of G means the minimum number of colours on edges which are incident to each vertex of G. We prove that if G is an edge-coloured graph with minimum colour degree at least 5, then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum colour degree at least 2. We show this theorem by proving amuch stronger form. Moreover, we point out an important relationship between our theorem and Bermond and Thomassen’s conjecture in digraphs.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"22 1","pages":"755 - 767"},"PeriodicalIF":0.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72794205","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 Ramsey numbers of hedgehogs","authors":"J. Fox, Ray Li","doi":"10.1017/S0963548319000312","DOIUrl":"https://doi.org/10.1017/S0963548319000312","url":null,"abstract":"Abstract The hedgehog Ht is a 3-uniform hypergraph on vertices $1, ldots ,t + left({matrix{t cr 2}}right)$ such that, for any pair (i, j) with 1 ≤ i < j ≤ t, there exists a unique vertex k > t such that {i, j, k} is an edge. Conlon, Fox and Rödl proved that the two-colour Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-colour Ramsey number grows exponentially in the square root of the number of vertices. They asked whether the two-colour Ramsey number of the hedgehog Ht is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r(Ht) = O(t2 ln t).","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"38 1","pages":"101 - 112"},"PeriodicalIF":0.0,"publicationDate":"2019-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85300625","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":"Robustness of randomized rumour spreading","authors":"Rami Daknama, K. Panagiotou, S. Reisser","doi":"10.1017/S0963548320000310","DOIUrl":"https://doi.org/10.1017/S0963548320000310","url":null,"abstract":"Abstract In this work we consider three well-studied broadcast protocols: push, pull and push&pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures. We explore in particular the following notion of local resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary may delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. On one hand, pull is robust with respect to all parameters that we consider. On the other hand, push may slow down significantly, even if the adversary may modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, push&pull is robust when no message transmission failures are considered, otherwise it may be slowed down. On the technical side, we develop two novel methods for the analysis of randomized rumour-spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemerédi’s Regularity Lemma.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"40 1","pages":"37 - 78"},"PeriodicalIF":0.0,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89403047","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":"Hamiltonicity in random directed graphs is born resilient","authors":"R. Montgomery","doi":"10.1017/S0963548320000140","DOIUrl":"https://doi.org/10.1017/S0963548320000140","url":null,"abstract":"Abstract Let \u0000${D_M}_{Mgeq 0}$\u0000 be the n-vertex random directed graph process, where \u0000$D_0$\u0000 is the empty directed graph on n vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each \u0000$$varepsilon > 0$$\u0000 , we show that, almost surely, any directed graph \u0000$D_M$\u0000 with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most \u0000$1/2-varepsilon$\u0000 of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is \u0000$(1/2-varepsilon)$\u0000 -resiliently Hamiltonian. Furthermore, for each \u0000$varepsilon > 0$\u0000 , we show that, almost surely, each directed graph \u0000$D_M$\u0000 in the sequence is not \u0000$(1/2+varepsilon)$\u0000 -resiliently Hamiltonian. This improves a result of Ferber, Nenadov, Noever, Peter and Škorić who showed, for each \u0000$varepsilon > 0$\u0000 , that the binomial random directed graph \u0000$D(n,p)$\u0000 is almost surely \u0000$(1/2-varepsilon)$\u0000 -resiliently Hamiltonian if \u0000$p=omega(log^8n/n)$\u0000 .","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"342 1","pages":"900 - 942"},"PeriodicalIF":0.0,"publicationDate":"2019-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79747510","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":"Triangle-degrees in graphs and tetrahedron coverings in 3-graphs","authors":"Victor Falgas‐Ravry, K. Markström, Yi Zhao","doi":"10.1017/S0963548320000061","DOIUrl":"https://doi.org/10.1017/S0963548320000061","url":null,"abstract":"Abstract We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c 1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree \u0000$delta_1(G)>d$\u0000 then every vertex of G is contained in a copy of F in G? We asymptotically determine c 1(n, F) when F is the generalized triangle \u0000$K_4^{(3)-}$\u0000 , and we give close to optimal bounds in the case where F is the tetrahedron \u0000$K_4^{(3)}$\u0000 (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with \u0000$m> n^2/4$\u0000 edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"24 1","pages":"175 - 199"},"PeriodicalIF":0.0,"publicationDate":"2019-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73877311","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":"Breaking bivariate records","authors":"J. A. Fill","doi":"10.1017/S0963548320000309","DOIUrl":"https://doi.org/10.1017/S0963548320000309","url":null,"abstract":"Abstract We establish a fundamental property of bivariate Pareto records for independent observations uniformly distributed in the unit square. We prove that the asymptotic conditional distribution of the number of records broken by an observation given that the observation sets a record is Geometric with parameter 1/2.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"52 1","pages":"105 - 123"},"PeriodicalIF":0.0,"publicationDate":"2019-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72579450","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":"Ramsey properties of randomly perturbed graphs: cliques and cycles","authors":"Shagnik Das, Andrew Treglown","doi":"10.1017/S0963548320000231","DOIUrl":"https://doi.org/10.1017/S0963548320000231","url":null,"abstract":"Abstract Given graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem. We also give a precise solution to the analogous question in the case when both H1 = Cs and H2 = Ct are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs, Kt) -Ramsey (for odd s ≽ 5 and t ≽ 4). To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"3 1","pages":"830 - 867"},"PeriodicalIF":0.0,"publicationDate":"2019-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89559490","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":"Finding independent transversals efficiently","authors":"Alessandra Graf, P. Haxell","doi":"10.1017/S0963548320000127","DOIUrl":"https://doi.org/10.1017/S0963548320000127","url":null,"abstract":"Abstract We give an efficient algorithm that, given a graph G and a partition V1,…,Vm of its vertex set, finds either an independent transversal (an independent set {v1,…,vm} in G such that ${v_i} in {V_i}$ for each i), or a subset ${cal B}$ of vertex classes such that the subgraph of G induced by $bigcupnolimits_{cal B}$ has a small dominating set. A non-algorithmic proof of this result has been known for a number of years and has been used to solve many other problems. Thus we are able to give algorithmic versions of many of these applications, a few of which we describe explicitly here.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"13 1","pages":"780 - 806"},"PeriodicalIF":0.0,"publicationDate":"2018-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74003361","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":"Large triangle packings and Tuza’s conjecture in sparse random graphs","authors":"Patrick Bennett, A. Dudek, Shira Zerbib","doi":"10.1017/S0963548320000115","DOIUrl":"https://doi.org/10.1017/S0963548320000115","url":null,"abstract":"Abstract The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"37 1","pages":"757 - 779"},"PeriodicalIF":0.0,"publicationDate":"2018-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81150371","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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