Independent Sets in Graphs with an Excluded Clique Minor

Let $G$ be a graph with $n$ vertices, with independence number $\alpha$, and with with no $K_{t+1}$-minor for some $t\geq5$. It is proved that $(2\alpha-1)(2t-5)\geq2n-5$

On Tree-Partition-Width

A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag in a tree-partition of $G$. An anonymous referee of the paper by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph with tree-width $k\geq3$ and maximum degree $\Delta\geq1$ has tree-partition-width at most $24k\Delta$. We prove that this bound is within a constant factor of optimal. In particular, for all $k\geq3$ and for all sufficiently large $\Delta$, we construct a graph with tree-width $k$, maximum degree $\Delta$, and tree-partition-width at least (\eighth-\epsilon)k\Delta. Moreover, we slightly improve the upper bound to ${5/2}(k+1)({7/2}\Delta-1)$ without the restriction that $k\geq3$

Drawing a Graph in a Hypercube

A $d$-dimensional hypercube drawing of a graph represents the vertices by distinct points in $\{0,1\}^d$, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections.Comment: Submitte

Air sampler collects and protects minute particles

Air ejector impactor sampler collects and protects samples of particles greater than 0.1 micron in diameter. In operation, it causes impaction of particle-laden air onto several collection surfaces within a collection cylinder. When not operating, the collector cylinder is maintained in a retracted state within a protective envelope

Attached Algae as an Indicator of Water Quality: A Study of the Viability of Genomic Taxonomic Methods

This research involved evaluating algae as an indicator of water quality in New Hampshire\u27s rivers, with a focus on the Great Bay Estuary. The project had three main goals. First, determining whether or not algae would work as an indicator of water quality in the great bay ecosystem, an environment where tidal currents are strong and water composition is mixed. The second goal was to compare traditional microscopic methods of taxonomy with emerging genomic methods, increasing the economic viability of attached algae monitoring. The third project goal, still underway, is to evaluate massive amounts of genomic data from the Great Bay ecosystem to see if other organisms might serve as viable indicators of environmental conditions in the bay. Despite the smalla tiny data set, traditional microscopic analysis results suggest that attached algae may be a viable indicator of water quality in the Great Bay Estuary. Further research including a larger data set will be required to evaluate the viability of genomic methods to supplement microscopic analyses, however early results have encouraged us to continue pursuing this research and expand our study to a larger portion of New Hampshire. Early results using bacteria have also encouraged us to continue analysis of genomic data in pursuit of goal number three, to find additional indicators that may serve as useful in water quality monitoring programs in New Hampshire

Colouring the Square of the Cartesian Product of Trees

We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree

The Physics Case for Extended Tevatron Running

Run II of the Tevatron collider at Fermilab is currently scheduled to end late in 2011. Given the current performance of the collider and of the CDF and D0 detectors, it is estimated that the current data set could be approximately doubled with a run extended into 2014. A few examples are presented of the physics potential of these additional statistics. These are discussed in the context of the expected reach of the LHC 7 TeV data and the existing Tevatron data. In particular, an extraordinary opportunity is described which could probe the existence of a standard model Higgs boson with mass in the currently preferred region between 115 GeV and 150 GeV.Comment: contribution to proceedings of HCP2010 - Toront