The Erdős-Ko-Rado properties of various graphs containing singletons

Let G=(V,E) be a graph. For r≥1, let be the family of independent vertex r-sets of G. For vV(G), let denote the star . G is said to be r-EKR if there exists vV(G) such that for any non-star family of pair-wise intersecting sets in . If the inequality is strict, then G is strictly r-EKR. Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if GΓ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2r is no larger than μ(G), the size of a smallest maximal independent vertex set of G, then G is r-EKR, and strictly so if 2r<μ(G). We show that in fact, if GΓ and 2r is no larger than the independence number of G, then G is r-EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ′Γ. We also confirm the conjecture for graphs in an even larger set Γ″Γ′

Compression and Erdos-Ko-Rado graphs

For a graph G and integer r >= 1 we denote the collection of independent r-setsof G by I^(r)(G). If v is in V(G) then I^(r)_v(G) is the collection of all independent r-sets containing v. A graph G is said to be r-EKR, for r >= 1, iff no intersecting family A of I^(r)(G) is larger than max_{v in V(G)} |I^(r)_v(G)|. There are various graphs that are known to have this property; the empty graph of order n >= 2r (this is the celebrated Erdos-Ko-Rado theorem), any disjoint union of atleast r copies of K_t for t >= 2, and any cycle. In this paper we show how these results can be extended to other classes of graphs via a compression proof technique. In particular we extend a theorem of Berge, showing that any disjoint union of at least r complete graphs, each of order at least two, is r-EKR. We also show that paths are r-EKR for all r >= 1

Colouring of cubic graphs by Steiner triple systems

Let S be a Steiner triple system and G a cubic graph. We say that G is S-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of S. We show that if S is a projective system PG(n, 2), n >= 2, then G is S-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an S-colouring for every Steiner triple system of order greater than 3. We establish a condition on a cubic graph with a bridge which ensures that it fails to have an S-colouring if S is an affine system, and we conjecture that this is the only obstruction to colouring any cubic graph with any non-projective system of order greater than 3

Graphs with the Erdos-Ko-Rado property

For a graph G, vertex v of G and integer r >= 1, we denote the family of independent r-sets of V(G) by I^(r)(G) and the subfamily {A in I^(r)(G): v in A} by I^(r)_v(G); such a family is called a star. Then, G is said to be r-EKR if no intersecting subfamily of I^(r)(G) is larger than the largest star in I^(r)(G). If every intersecting subfamily of I^(r)_v(G) of maximum size is a star, then G is said to be strictly r-EKR. We show that if a graph is r-EKR then its lexicographic product with any complete graph is r-EKR. For any graph G, we define mu(G) to be the minimum size of a maximal independent vertex set. We conjecture that, if 1 <= r <= 1/2 mu(G), then G is r-EKR, and if r < 1/2 mu(G), then G is strictly r-EKR. This is known to be true when G is an empty graph, a cycle, a path or the disjoint union of complete graphs. We show that it is also true when G is the disjoint union of a pair of complete multipartite graphs

Munc18-Bound Syntaxin Readily Forms SNARE Complexes with Synaptobrevin in Native Plasma Membranes

Munc18–1, a protein essential for regulated exocytosis in neurons and neuroendocrine cells, belongs to the family of Sec1/Munc18-like (SM) proteins. In vitro, Munc18–1 forms a tight complex with the SNARE syntaxin 1, in which syntaxin is stabilized in a closed conformation. Since closed syntaxin is unable to interact with its partner SNAREs SNAP-25 and synaptobrevin as required for membrane fusion, it has hitherto not been possible to reconcile binding of Munc18–1 to syntaxin 1 with its biological function. We now show that in intact and exocytosis-competent lawns of plasma membrane, Munc18–1 forms a complex with syntaxin that allows formation of SNARE complexes. Munc18–1 associated with membrane-bound syntaxin 1 can be effectively displaced by adding recombinant synaptobrevin but not syntaxin 1 or SNAP-25. Displacement requires the presence of endogenous SNAP-25 since no displacement is observed when chromaffin cell membranes from SNAP-25–deficient mice are used. We conclude that Munc18–1 allows for the formation of a complex between syntaxin and SNAP-25 that serves as an acceptor for vesicle-bound synaptobrevin and that thus represents an intermediate in the pathway towards exocytosis

Genome-wide analysis of Schistosoma mansoni reveals limited population structure and possible praziquantel drug selection pressure within Ugandan hot-spot communities

Populations within schistosomiasis control areas, especially those in Africa, are recommended to receive regular mass drug administration (MDA) with praziquantel (PZQ) as the main strategy for controlling the disease. The impact of PZQ treatment on schistosome genetics remains poorly understood, and is limited by a lack of high-resolution genetic data on the population structure of parasites within these control areas. We generated whole-genome sequence data from 174 individual miracidia collected from both children and adults from fishing communities on islands in Lake Victoria in Uganda that had received either annual or quarterly MDA with PZQ over four years, including samples collected immediately before and four weeks after treatment. Genome variation within and between samples was characterised and we investigated genomic signatures of natural selection acting on these populations that could be due to PZQ treatment. The parasite population on these islands was more diverse than found in nearby villages on the lake shore. We saw little or no genetic differentiation between villages, or between the groups of villages with different treatment intensity, but slightly higher genetic diversity within the pre-treatment compared to post-treatment parasite populations. We identified classes of genes significantly enriched within regions of the genome with evidence of recent positive selection among post-treatment and intensively treated parasite populations. The differential selection observed in post-treatment and pre-treatment parasite populations could be linked to any reduced susceptibility of parasites to praziquantel treatment

Finishing the euchromatic sequence of the human genome

The sequence of the human genome encodes the genetic instructions for human physiology, as well as rich information about human evolution. In 2001, the International Human Genome Sequencing Consortium reported a draft sequence of the euchromatic portion of the human genome. Since then, the international collaboration has worked to convert this draft into a genome sequence with high accuracy and nearly complete coverage. Here, we report the result of this finishing process. The current genome sequence (Build 35) contains 2.85 billion nucleotides interrupted by only 341 gaps. It covers ∼99% of the euchromatic genome and is accurate to an error rate of ∼1 event per 100,000 bases. Many of the remaining euchromatic gaps are associated with segmental duplications and will require focused work with new methods. The near-complete sequence, the first for a vertebrate, greatly improves the precision of biological analyses of the human genome including studies of gene number, birth and death. Notably, the human enome seems to encode only 20,000-25,000 protein-coding genes. The genome sequence reported here should serve as a firm foundation for biomedical research in the decades ahead

Overlap colourings and homomorphisms of graphs

An (r, λ) overlap colouring of a graph G has r colours at each vertex, any two adjacent vertices sharing exactly λ colours. A theory analogous to multichromatic and fractional chromatic theory is developed. In particular, all the overlap chromatic numbers of cycle graphs are computed. It is shown that if a graph G contains an odd cycle C2p+1 and has the same p-fold chromatic number as C2p+1, then all its overlap chromatic numbers are the same as those of C2p+1. The core of a graph is the smallest induced subgraph to which it has a homomorphism. It is shown that some pairs of graphs with the same multichromatic numbers have different sets of overlap chromatic numbers, and that some graphs with non-isomorphic cores have the same sets of overlap chromatic numbers. (In particular, any non-bipartite series-parallel graph has the same overlap chromatic as its smallest odd cycle.) Thus classifying graphs by overlap chromatic properties is intermediate between classifying them by multichromatic properties and classifying them by cores