94 research outputs found

### An asymptotic existence result on compressed sensing matrices

For any rational number $h$ and all sufficiently large $n$ we give a
deterministic construction for an $n\times \lfloor hn\rfloor$ compressed
sensing matrix with $(\ell_1,t)$-recoverability where $t=O(\sqrt{n})$. Our
method uses pairwise balanced designs and complex Hadamard matrices in the
construction of $\epsilon$-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 $(\ell_1,t)$-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

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

### On Hamilton decompositions of infinite circulant graphs

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

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

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

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

We show that the complete graph on $n$ vertices can be decomposed into $t$
cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is odd, $3\leq
m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\cdots+m_t=\binom n2$. We also show
that the complete graph on $n$ vertices can be decomposed into a perfect
matching and $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if
$n$ is even, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\ldots+m_t=\binom
n2-\frac n2$.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

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