488 research outputs found

    A formulation of a (q+1,8)-cage

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    Let q2q\ge 2 be a prime power. In this note we present a formulation for obtaining the known (q+1,8)(q+1,8)-cages which has allowed us to construct small (k,g)(k,g)--graphs for k=q1,qk=q-1, q and g=7,8g=7,8. Furthermore, we also obtain smaller (q,8)(q,8)-graphs for even prime power qq.Comment: 14 pages, 2 figure

    Cooperative Local Repair in Distributed Storage

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    Erasure-correcting codes, that support local repair of codeword symbols, have attracted substantial attention recently for their application in distributed storage systems. This paper investigates a generalization of the usual locally repairable codes. In particular, this paper studies a class of codes with the following property: any small set of codeword symbols can be reconstructed (repaired) from a small number of other symbols. This is referred to as cooperative local repair. The main contribution of this paper is bounds on the trade-off of the minimum distance and the dimension of such codes, as well as explicit constructions of families of codes that enable cooperative local repair. Some other results regarding cooperative local repair are also presented, including an analysis for the well-known Hadamard/Simplex codes.Comment: Fixed some minor issues in Theorem 1, EURASIP Journal on Advances in Signal Processing, December 201

    A bound on the number of edges in graphs without an even cycle

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    We show that, for each fixed kk, an nn-vertex graph not containing a cycle of length 2k2k has at most 80klogkn1+1/k+O(n)80\sqrt{k}\log k\cdot n^{1+1/k}+O(n) edges.Comment: 16 pages, v2 appeared in Comb. Probab. Comp., v3 fixes an error in v2 and explains why the method in the paper cannot improve the power of k further, v4 fixes the proof of Theorem 12 introduced in v

    The history of degenerate (bipartite) extremal graph problems

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    This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

    Families of Small Regular Graphs of Girth 5

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    In this paper we obtain (q+3)(q+3)--regular graphs of girth 5 with fewer vertices than previously known ones for q=13,17,19q=13,17,19 and for any prime q23q \ge 23 performing operations of reductions and amalgams on the Levi graph BqB_q of an elliptic semiplane of type C{\cal C}. We also obtain a 13-regular graph of girth 5 on 236 vertices from B11B_{11} using the same technique

    Approximating the Largest Root and Applications to Interlacing Families

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    We study the problem of approximating the largest root of a real-rooted polynomial of degree nn using its top kk coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in kk that use the top kk coefficients to approximate the maximum root within a factor of n1/kn^{1/k} and 1+O(lognk)21+O(\tfrac{\log n}{k})^2 when klognk\leq \log n and k>lognk>\log n respectively. We also prove corresponding information-theoretic lower bounds of nΩ(1/k)n^{\Omega(1/k)} and 1+Ω(log2nkk)21+\Omega\left(\frac{\log \frac{2n}{k}}{k}\right)^2, and show strong lower bounds for noisy version of the problem in which one is given access to approximate coefficients. This problem has applications in the context of the method of interlacing families of polynomials, which was used for proving the existence of Ramanujan graphs of all degrees, the solution of the Kadison-Singer problem, and bounding the integrality gap of the asymmetric traveling salesman problem. All of these involve computing the maximum root of certain real-rooted polynomials for which the top few coefficients are accessible in subexponential time. Our results yield an algorithm with the running time of 2O~(n3)2^{\tilde O(\sqrt[3]n)} for all of them
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