A sequence of triangle-free pseudorandom graphs

A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs.Comment: 6 page

Viruses Associated with Ovarian Degeneration in Apis mellifera L. Queens

Queen fecundity is a critical issue for the health of honeybee (Apis mellifera L.) colonies, as she is the only reproductive female in the colony and responsible for the constant renewal of the worker bee population. Any factor affecting the queen's fecundity will stagnate colony development, increasing its susceptibility to opportunistic pathogens. We discovered a pathology affecting the ovaries, characterized by a yellow discoloration concentrated in the apex of the ovaries resulting from degenerative lesions in the follicles. In extreme cases, marked by intense discoloration, the majority of the ovarioles were affected and these cases were universally associated with egg-laying deficiencies in the queens. Microscopic examination of the degenerated follicles showed extensive paracrystal lattices of 30 nm icosahedral viral particles. A cDNA library from degenerated ovaries contained a high frequency of deformed wing virus (DWV) and Varroa destructor virus 1 (VDV-1) sequences, two common and closely related honeybee Iflaviruses. These could also be identified by in situ hybridization in various parts of the ovary. A large-scale survey for 10 distinct honeybee viruses showed that DWV and VDV-1 were by far the most prevalent honeybee viruses in queen populations, with distinctly higher prevalence in mated queens (100% and 67%, respectively for DWV and VDV-1) than in virgin queens (37% and 0%, respectively). Since very high viral titres could be recorded in the ovaries and abdomens of both functional and deficient queens, no significant correlation could be made between viral titre and ovarian degeneration or egg-laying deficiency among the wider population of queens. Although our data suggest that DWV and VDV-1 have a role in extreme cases of ovarian degeneration, infection of the ovaries by these viruses does not necessarily result in ovarian degeneration, even at high titres, and additional factors are likely to be involved in this pathology

How unproportional must a graph be?

Let $u_k(G,p)$ be the maximum over all $k$-vertex graphs $F$ of by how much the number of induced copies of $F$ in $G$ differs from its expectation in the binomial random graph with the same number of vertices as $G$ and with edge probability $p$. This may be viewed as a measure of how close $G$ is to being $p$-quasirandom. For a positive integer $n$ and $0<p<1$, let $D(n,p)$ be the distance from $p\binom{n}{2}$ to the nearest integer. Our main result is that, for fixed $k\ge 4$ and for $n$ large, the minimum of $u_k(G,p)$ over $n$-vertex graphs has order of magnitude $\Theta\big(\max\{D(n,p), p(1-p)\} n^{k-2}\big)$ provided that $p(1-p)n^{1/2} \to \infty$

Intersections of hypergraphs

Given two weighted k-uniform hypergraphs G, H of order n, how much (or little) can we make them overlap by placing them on the same vertex set? If we place them at random, how concentrated is the distribution of the intersection? The aim of this paper is to investigate these questions