352 research outputs found

    Pseudo-random graphs

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    Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate. Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs and the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page

    Local statistics of lattice dimers

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    We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy μ\mu on the space of tilings of the plane with dominos. We construct a measure ν\nu on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the ν\nu-measures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for μ\mu and ν\nu, and compute the rate of convergence to mixing (the correlation between distant events). For the measure ν\nu we compute the variance of the height function.Comment: 27 pages, 6 figure

    Ramsey-nice families of graphs

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    For a finite family F\mathcal{F} of fixed graphs let Rk(F)R_k(\mathcal{F}) be the smallest integer nn for which every kk-coloring of the edges of the complete graph KnK_n yields a monochromatic copy of some F∈FF\in\mathcal{F}. We say that F\mathcal{F} is kk-nice if for every graph GG with χ(G)=Rk(F)\chi(G)=R_k(\mathcal{F}) and for every kk-coloring of E(G)E(G) there exists a monochromatic copy of some F∈FF\in\mathcal{F}. It is easy to see that if F\mathcal{F} contains no forest, then it is not kk-nice for any kk. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs F\mathcal{F} that contains at least one forest, and for all k≥k0(F)k\geq k_0(\mathcal{F}) (or at least for infinitely many values of kk), F\mathcal{F} is kk-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family F\mathcal{F} containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

    The asymptotic determinant of the discrete Laplacian

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    We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R^2. As an application of this result, we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.Comment: 36 pages, 4 figures, to appear in Acta Mathematic

    Polynomial-time perfect matchings in dense hypergraphs

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    Let HH be a kk-graph on nn vertices, with minimum codegree at least n/k+cnn/k + cn for some fixed c>0c > 0. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in HH or a certificate that none exists. This essentially solves a problem of Karpi\'nski, Ruci\'nski and Szyma\'nska; Szyma\'nska previously showed that this problem is NP-hard for a minimum codegree of n/k−cnn/k - cn. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.Comment: 64 pages. Update includes minor revisions. To appear in Advances in Mathematic
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