An immersion of a graph H into a graph G is a one-to-one mapping f : V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P uv corresponding to edge uv has endpoints f (u) and f (v). The immersion is strong if the paths P uv are internally disjoint from f (V (H)). It is proved that for every positive integer t, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd 3/2 , where c > 0 is an absolute constant. For small values of t, 1 ≤ t ≤ 7, every simple graph of minimum degree at least t − 1 contains an immersion of K t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

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Recent data suggest that soluble, non-fibrillar assemblies of the amyloid β-protein (Aβ) may mediate the synaptic deficits that characterize the early stages of Alzheimer's disease. Consequently, much effort has been expended in isolating and studying a variety of different Aβ assemblies. Here, we describe the use of immunoprecipitation/western blotting and size exclusion chromatography/western blotting to characterize Aβ present in conditioned medium from cultured cells, human cerebrospinal fluid, and human cortex extracted with aqueous buffer, detergent, and formic acid.

Although plaques composed of the amyloid β-protein (Aβ) are considered a defining feature of Alzheimer's disease (AD), they are also found in cognitively normal individuals and extensive evidence suggests that non-plaque, water-soluble forms of Aβ may play a role in AD pathogenesis. However, the relationship between the levels of water-soluble Aβ and the clinical severity of disease has never been investigated. Here, we present results of a pilot study designed to examine the levels of water-soluble forms of Aβ in brains of individuals who died at clinically distinct stages of AD. Using a serial extraction method, we also investigated the levels of triton-soluble and formic acid-soluble Aβ. We found that water-soluble and detergent-soluble Aβ monomer and SDS-stable dimer were elevated in AD and that the levels of water soluble Aβ did not increase with plaque pathology. These results support the notion that both water- and detergent-soluble Aβ are important in AD and are not simply released from plaques by mechanical disruption. Moreover, the fact that the levels of water- and triton-soluble Aβ were similar in very mild/mild AD and moderate/severe AD suggests that once a certain level of these species is attained, further accumulation is not necessary for the disease to progress. Consequently, therapeutic targeting of water-soluble Aβ should best benefit individuals in earliest phases of the disease process.

Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure theorem for graphs without K timmersions; it is based on the Gomory-Hu theorem. The same proof also works to establish a rough structure theorem for Eulerian digraphs without K t -immersions, where K t denotes the bidirected complete digraph of order t. * mdevos@sfu.ca. Supported in part by an NSERC Discovery Grant (Canada) and a Sloan Fellowship. † jessica mcdonald@sfu.ca. Supported by an NSERC Postdoctoral Fellowship (Canada) ‡ mohar@sfu.ca.