2,801 research outputs found

    Maximum Matching in Turnstile Streams

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    We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass 22-approximation streaming algorithm can be easily obtained with space O(nlogn)O(n \log n), where nn denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space O(n3/2δ)O(n^{3/2-\delta}) is granted, for every δ>0\delta > 0. Specifically, for every 0ϵ10 \le \epsilon \le 1, we show that in the one-pass turnstile streaming model, in order to compute a O(nϵ)O(n^{\epsilon})-approximation, space Ω(n3/24ϵ)\Omega(n^{3/2 - 4\epsilon}) is required for constant error randomized algorithms, and, up to logarithmic factors, space O(n22ϵ)O( n^{2-2\epsilon} ) is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest

    Some colouring problems for Paley graphs

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    The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved

    Percolation on sparse random graphs with given degree sequence

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    We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards we focus on site percolation where the vertices are retained with probability p. We establish critical values for p above which a giant component emerges in both cases. Moreover, we show that in fact these coincide. As a special case, our results apply to power law random graphs. We obtain rigorous proofs for formulas derived by several physicists for such graphs.Comment: 20 page

    Factors of IID on Trees

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    Classical ergodic theory for integer-group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We present some illustrative results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. Our results include bounds on minimum and maximum bisection for random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur

    On the expected number of perfect matchings in cubic planar graphs

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    A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with nn vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically cγnc\gamma^n, where c>0c>0 and γ1.14196\gamma \sim 1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.Comment: 19 pages, 4 figure
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