346,668 research outputs found
Largest sparse subgraphs of random graphs
For the Erd\H{o}s-R\'enyi random graph G(n,p), we give a precise asymptotic
formula for the size of a largest vertex subset in G(n,p) that induces a
subgraph with average degree at most t, provided that p = p(n) is not too small
and t = t(n) is not too large. In the case of fixed t and p, we find that this
value is asymptotically almost surely concentrated on at most two explicitly
given points. This generalises a result on the independence number of random
graphs. For both the upper and lower bounds, we rely on large deviations
inequalities for the binomial distribution.Comment: 15 page
The structure of spray-dried detergent powders
The complex multi-scale structure of spray-dried detergent granules has been characterized using a range of techniques including microscopy, wide-angle and small-angle X-ray scattering and X-ray microtomography. Four simple model formulations based on linear alkyl benzene sulphonate (NaLAS) and sodium sulphate were used to probe the influence of initial slurry water content and sodium silicate on the structure. The structure can be viewed as a porous matrix consisting of liquid crystalline NaLAS, sodium sulphate and binder in which large crystals of sodium sulphate are embedded. These large crystals were initially undissolved in the slurry and are consequently reduced in number in the product made from higher water content slurry. The both slurry water content and sodium silicate changed the polymorphs, and the d-spacing of the lamellae. The surface micro-structure and particle morphology can also be significantly affected with the high initial water content; particles having a distinct agglomerated and blistered structure
Saturation numbers for Ramsey-minimal graphs
Given graphs , a graph is -Ramsey-minimal if every -coloring of the edges of contains a
monochromatic in color for some , but any proper
subgraph of does not possess this property. We define
to be the family of -Ramsey-minimal graphs. A graph is \dfn{-saturated} if no element of
is a subgraph of , but for any edge in , some element of
is a subgraph of . We define
to be the minimum number of edges
over all -saturated graphs on
vertices. In 1987, Hanson and Toft conjectured that for , where is the classical
Ramsey number for complete graphs. The first non-trivial case of Hanson and
Toft's conjecture for sufficiently large was setteled in 2011, and is so
far the only settled case. Motivated by Hanson and Toft's conjecture, we study
the minimum number of edges over all -saturated graphs on vertices, where is the
family of all trees on vertices. We show that for , . For and , we obtain an
asymptotic bound for .Comment: to appear in Discrete Mathematic
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