{"title":"Maker–Breaker percolation games I: crossing grids","authors":"A. Day, Victor Falgas‐Ravry","doi":"10.1017/S0963548320000097","DOIUrl":"https://doi.org/10.1017/S0963548320000097","url":null,"abstract":"Abstract Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n. Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition. If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ. If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m. Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"17 1","pages":"200 - 227"},"PeriodicalIF":0.0,"publicationDate":"2018-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80745045","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":"Improved Ramsey-type results for comparability graphs","authors":"D'aniel Kor'andi, István Tomon","doi":"10.1017/S0963548320000103","DOIUrl":"https://doi.org/10.1017/S0963548320000103","url":null,"abstract":"Abstract Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size \u0000$n^{1/(r+1)}$\u0000. This bound is known to be tight for \u0000$r=1$\u0000. The question whether it is optimal for \u0000$rge 2$\u0000 was studied by Dumitrescu and Tóth. We prove that it is essentially best possible for \u0000$r=2$\u0000, as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size \u0000$n^{1/3+o(1)}$\u0000. Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G nor its complement contain a complete bipartite graph with parts of size \u0000$cn/{(log n)^r}$\u0000. With this, we improve a result of Fox and Pach.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"5 1","pages":"747 - 756"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87094387","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 minimal Ramsey graphs and Ramsey equivalence in multiple colours","authors":"Dennis Clemens, Anita Liebenau, D. Reding","doi":"10.1017/S0963548320000036","DOIUrl":"https://doi.org/10.1017/S0963548320000036","url":null,"abstract":"Abstract For an integer q ⩾ 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H yet no proper subgraph of G has this property, then G is called q-Ramsey-minimal for H. Generalizing a statement by Burr, Nešetřil and Rödl from 1977, we prove that, for q ⩾ 3, if G is a graph that is not q-Ramsey for some graph H, then G is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H as long as H is 3-connected or isomorphic to the triangle. For such H, the following are some consequences. For 2 ⩽ r < q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H. For every q ⩾ 3, there are q-Ramsey-minimal graphs for H of arbitrarily large maximum degree, genus and chromatic number. The collection ${mathcal M_q(H) colon H text{ is 3-connected or } K_3}$ forms an antichain with respect to the subset relation, where $mathcal M_q(H)$ denotes the set of all graphs that are q-Ramsey-minimal for H. We also address the question of which pairs of graphs satisfy $mathcal M_q(H_1)=mathcal M_q(H_2)$ , in which case H 1 and H 2 are called q-equivalent. We show that two graphs H 1 and H 2 are q-equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q ⩾ 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"102 1","pages":"537 - 554"},"PeriodicalIF":0.0,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75794684","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}
Matija Bucić, Sven Heberle, Shoham Letzter, B. Sudakov
{"title":"Monochromatic trees in random tournaments","authors":"Matija Bucić, Sven Heberle, Shoham Letzter, B. Sudakov","doi":"10.1017/S0963548319000373","DOIUrl":"https://doi.org/10.1017/S0963548319000373","url":null,"abstract":"Abstract We prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order $O(n{rm{/}}sqrt {{rm{log}} n} )$. This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"46 1","pages":"318 - 345"},"PeriodicalIF":0.0,"publicationDate":"2018-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78469294","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":"Discrete derivative asymptotics of the β-Hermite eigenvalues","authors":"Gopal K. Goel, Andrew Ahn","doi":"10.1017/S0963548319000087","DOIUrl":"https://doi.org/10.1017/S0963548319000087","url":null,"abstract":"Abstract We consider the asymptotics of the difference between the empirical measures of the β-Hermite tridiagonal matrix and its minor. We prove that this difference has a deterministic limit and Gaussian fluctuations. Through a correspondence between measures and continual Young diagrams, this deterministic limit is identified with the Vershik–Kerov–Logan–Shepp curve. Moreover, the Gaussian fluctuations are identified with a sectional derivative of the Gaussian free field.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"56 1","pages":"657 - 674"},"PeriodicalIF":0.0,"publicationDate":"2018-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86213177","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":"FKN theorem for the multislice, with applications","authors":"Yuval Filmus","doi":"10.1017/S0963548319000361","DOIUrl":"https://doi.org/10.1017/S0963548319000361","url":null,"abstract":"Abstract The Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice. As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"5 1","pages":"200 - 212"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74763408","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":"Hamiltonian Berge cycles in random hypergraphs","authors":"Deepak Bal, R. Berkowitz, Pat Devlin, M. Schacht","doi":"10.1017/S0963548320000437","DOIUrl":"https://doi.org/10.1017/S0963548320000437","url":null,"abstract":"Abstract In this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For \u0000$rgeq 3$\u0000 we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r-graph, and we also show that the 2-out random r-graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"2 1","pages":"228 - 238"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83556410","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":"Approximately counting bases of bicircular matroids","authors":"Heng Guo, M. Jerrum","doi":"10.1017/S0963548320000292","DOIUrl":"https://doi.org/10.1017/S0963548320000292","url":null,"abstract":"Abstract We give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"6 1","pages":"124 - 135"},"PeriodicalIF":0.0,"publicationDate":"2018-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89067354","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}
Beka Ergemlidze, E. Györi, Abhishek Methuku, Nika Salia, C. Tompkins, Oscar Zamora
{"title":"Avoiding long Berge cycles: the missing cases k = r + 1 and k = r + 2","authors":"Beka Ergemlidze, E. Györi, Abhishek Methuku, Nika Salia, C. Tompkins, Oscar Zamora","doi":"10.1017/S0963548319000415","DOIUrl":"https://doi.org/10.1017/S0963548319000415","url":null,"abstract":"Abstract The maximum size of an r-uniform hypergraph without a Berge cycle of length at least k has been determined for all k ≥ r + 3 by Füredi, Kostochka and Luo and for k < r (and k = r, asymptotically) by Kostochka and Luo. In this paper we settle the remaining cases: k = r + 1 and k = r + 2, proving a conjecture of Füredi, Kostochka and Luo.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"80 4 1","pages":"423 - 435"},"PeriodicalIF":0.0,"publicationDate":"2018-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77402173","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":"Long Monotone Trails in Random Edge-Labellings of Random Graphs","authors":"Omer Angel, Asaf Ferber, B. Sudakov, V. Tassion","doi":"10.1017/S096354831900018X","DOIUrl":"https://doi.org/10.1017/S096354831900018X","url":null,"abstract":"Abstract Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n-1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path. In this paper we consider the random graph G = Gn,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average-case version of the result of Graham and Kleitman, showing that the random edge ordering of Kn has w.h.p. an increasing trail of length (1-o(1))en, and that this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erdős-Renyi graphs with p = o(1).","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":"30 6 1","pages":"22 - 30"},"PeriodicalIF":0.0,"publicationDate":"2018-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83001807","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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