5,980 research outputs found

    Universal graphs with a forbidden subtree

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    We show that the problem of the existence of universal graphs with specified forbidden subgraphs can be systematically reduced to certain critical cases by a simple pruning technique which simplifies the underlying structure of the forbidden graphs, viewed as trees of blocks. As an application, we characterize the trees T for which a universal countable T-free graph exists

    The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane

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    We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio

    Random induced subgraphs of Cayley graphs induced by transpositions

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    In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, λn\lambda_n. Our main result is that for any minimal generating set of transpositions, for probabilities λn=1+Ï”nn−1\lambda_n=\frac{1+\epsilon_n}{n-1} where n−1/3+Ύ≀ϔn0n^{-{1/3}+\delta}\le \epsilon_n0, a random induced subgraph has a.s. a unique largest component of size ℘(Ï”n)1+Ï”nn−1n!\wp(\epsilon_n)\frac{1+\epsilon_n}{n-1}n!, where ℘(Ï”n)\wp(\epsilon_n) is the survival probability of a specific branching process.Comment: 18 pages, 1 figur

    The time of graph bootstrap percolation

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    Graph bootstrap percolation, introduced by Bollob\'as in 1968, is a cellular automaton defined as follows. Given a "small" graph HH and a "large" graph G=G0⊆KnG = G_0 \subseteq K_n, in consecutive steps we obtain Gt+1G_{t+1} from GtG_t by adding to it all new edges ee such that GtâˆȘeG_t \cup e contains a new copy of HH. We say that GG percolates if for some t≄0t \geq 0, we have Gt=KnG_t = K_n. For H=KrH = K_r, the question about the size of the smallest percolating graphs was independently answered by Alon, Frankl and Kalai in the 1980's. Recently, Balogh, Bollob\'as and Morris considered graph bootstrap percolation for G=G(n,p)G = G(n,p) and studied the critical probability pc(n,Kr)p_c(n,K_r), for the event that the graph percolates with high probability. In this paper, using the same setup, we determine, up to a logarithmic factor, the critical probability for percolation by time tt for all 1≀t≀Clog⁥log⁥n1 \leq t \leq C \log\log n.Comment: 18 pages, 3 figure

    Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings

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    Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen's theorem and of Strojohann's determinant algorithm. It allows us to give new and simple solutions to the following problems: * Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^{\omega}) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with non-negative weights) this matches the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the other hand, no algorithm working in \tilde{O}(Wn^{\omega}) time was previously known for undirected graphs with negative weights. Furthermore our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. * Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^{\omega}) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge no algorithm with this running time was known for undirected graphs with negative weights. * Finding Minimum Weight Perfect Matchings -- We present an \tilde{O}(Wn^{\omega}) time algorithm for finding minimum weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski in 2006, who presented such an algorithm but only in the case of bipartite graphs. In order to solve minimum weight perfect matching problem we develop a novel combinatorial interpretation of the dual solution which sheds new light on this problem. Such a combinatorial interpretation was not know previously, and is of independent interest.Comment: To appear in FOCS 201
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