94 research outputs found

    An asymptotic existence result on compressed sensing matrices

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    For any rational number hh and all sufficiently large nn we give a deterministic construction for an n×hnn\times \lfloor hn\rfloor compressed sensing matrix with (1,t)(\ell_1,t)-recoverability where t=O(n)t=O(\sqrt{n}). Our method uses pairwise balanced designs and complex Hadamard matrices in the construction of ϵ\epsilon-equiangular frames, which we introduce as a generalisation of equiangular tight frames. The method is general and produces good compressed sensing matrices from any appropriately chosen pairwise balanced design. The (1,t)(\ell_1,t)-recoverability performance is specified as a simple function of the parameters of the design. To obtain our asymptotic existence result we prove new results on the existence of pairwise balanced designs in which the numbers of blocks of each size are specified.Comment: 15 pages, no figures. Minor improvements and updates in February 201

    On Hamilton Decompositions of Line Graphs of Non-Hamiltonian Graphs and Graphs without Separating Transitions

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    In contrast with Kotzig's result that the line graph of a 33-regular graph XX is Hamilton decomposable if and only if XX is Hamiltonian, we show that for each integer k4k\geq 4 there exists a simple non-Hamiltonian kk-regular graph whose line graph has a Hamilton decomposition. We also answer a question of Jackson by showing that for each integer k3k\geq 3 there exists a simple connected kk-regular graph with no separating transitions whose line graph has no Hamilton decomposition

    On Hamilton decompositions of infinite circulant graphs

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    The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}

    Compressed sensing with combinatorial designs: theory and simulations

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    In 'An asymptotic result on compressed sensing matrices', a new construction for compressed sensing matrices using combinatorial design theory was introduced. In this paper, we use deterministic and probabilistic methods to analyse the performance of matrices obtained from this construction. We provide new theoretical results and detailed simulations. These simulations indicate that the construction is competitive with Gaussian random matrices, and that recovery is tolerant to noise. A new recovery algorithm tailored to the construction is also given.Comment: 18 pages, 3 figure

    Decomposing graphs of high minimum degree into 4-cycles

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    If a graph G decomposes into edge-disjoint 4-cycles, then each vertex of G has even degree and 4 divides the number of edges in G. It is shown that these obvious necessary conditions are also sufficient when G is any simple graph having minimum degree at least (3132+on(1))n, where n is the number of vertices in G. This improves the bound given by Gustavsson (PhD Thesis, University of Stockholm, 1991), who showed (as part of a more general result) sufficiency for simple graphs with minimum degree at least (1-10-94+on(1))n. On the other hand, it is known that for arbitrarily large values of n there exist simple graphs satisfying the obvious necessary conditions, having n vertices and minimum degree 35n-1, but having no decomposition into edge-disjoint 4-cycles. We also show that if G is a bipartite simple graph with n vertices in each part, then the obvious necessary conditions for G to decompose into 4-cycles are sufficient when G has minimum degree at least (3132+on(1))n

    Completing partial Latin squares with two filled rows and two filled columns

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    It is shown that any partial Latin square of order at least six which consists of two filled rows and two filled columns can be completed

    Decompositions of complete graphs into cycles of arbitrary lengths

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    We show that the complete graph on nn vertices can be decomposed into tt cycles of specified lengths m1,,mtm_1,\ldots,m_t if and only if nn is odd, 3min3\leq m_i\leq n for i=1,,ti=1,\ldots,t, and m1++mt=(n2)m_1+\cdots+m_t=\binom n2. We also show that the complete graph on nn vertices can be decomposed into a perfect matching and tt cycles of specified lengths m1,,mtm_1,\ldots,m_t if and only if nn is even, 3min3\leq m_i\leq n for i=1,,ti=1,\ldots,t, and m1++mt=(n2)n2m_1+\ldots+m_t=\binom n2-\frac n2.Comment: 182 pages, 0 figures, A condensed version of this paper was published as "Cycle decompositions V: Complete graphs into cycles of arbitrary lengths" (see reference [24]). Here, we include supplementary data and some proofs which were omitted from that pape