Topics in graph colouring and graph structures

This thesis investigates problems in a number of different areas of graph theory. These problems are related in the sense that they mostly concern the colouring or structure of the underlying graph. The first problem we consider is in Ramsey Theory, a branch of graph theory stemming from the eponymous theorem which, in its simplest form, states that any sufficiently large graph will contain a clique or anti-clique of a specified size. The problem of finding the minimum size of underlying graph which will guarantee such a clique or anti-clique is an interesting problem in its own right, which has received much interest over the last eighty years but which is notoriously intractable. We consider a generalisation of this problem. Rather than edges being present or not present in the underlying graph, each is assigned one of three possible colours and, rather than considering cliques, we consider cycles. Combining regularity and stability methods, we prove an exact result for a triple of long cycles. We then move on to consider removal lemmas. The classic Removal Lemma states that, for n sufficiently large, any graph on n vertices containing o(n^3) triangles can be made triangle-free by the removal of o(n^2) edges. Utilising a coloured hypergraph generalisation of this result, we prove removal lemmas for two classes of multinomials. Next, we consider a problem in fractional colouring. Since finding the chromatic number of a given graph can be viewed as an integer programming problem, it is natural to consider the solution to the corresponding linear programming problem. The solution to this LP-relaxation is called the fractional chromatic number. By a probabilistic method, we improve on the best previously known bound for the fractional chromatic number of a triangle-free graph with maximum degree at most three. Finally, we prove a weak version of Vizing's Theorem for hypergraphs. We prove that, if H is an intersecting 3-uniform hypergraph with maximum degree D and maximum multiplicity m, then H has at most 2D+m edges. Furthermore, we prove that the unique structure achieving this maximum is m copies of the Fano Plane

Village polytechnics in central Kenya : progress, problems and prospects

This paper gives the overall findings of a study designed to assess the performance of village polytechnics in Central Kenya in terms of the promotion of rural development, the attitude of trainees to employment prospects in the rural areas and the city and the extent to which the success of village polytechnics is dependent on the standard of living in the rural areas. In the sample of 23 VPs, most had been successful in placing their leavers in employment in their local areas although no consistent monitoring of leavers had been done at the time of the survey. Although most of the 175 trainees interviewed expressed a desire to work in their home area a sizeable minority expressed a preference to work in a town, particularly Nairobi, although the general image of Nairobi is a vague one. The role of VPs in promoting rural development is seen as over-ambitious since the success of the VPs tends to depend highly on the pre-existing levels of rural prospects. Several problems restricting the development of VPs are identified and solutions suggested. The greatest danger to the development of VPs appears to be the growing lack of flexibility in organization which prevents VPs from reacting quickly enough to changes in the local economy

The fractional chromatic number of triangle-free subcubic graphs

Heckman and Thomas conjectured that the fractional chromatic number of any triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami and Zhu and of Lu and Peng, we prove that the fractional chromatic number of any triangle-free subcubic graph is at most 32/11 (which is roughly 2.909)

Making Sense of Listening: The IMAP Test Battery

The ability to hear is only the first step towards making sense of the range of information contained in an auditory signal. Of equal importance are the abilities to extract and use the information encoded in the auditory signal. We refer to these as listening skills (or auditory processing AP). Deficits in these skills are associated with delayed language and literacy development, though the nature of the relevant deficits and their causal connection with these delays is hotly debated

The NextGen Model Atmosphere grid: II. Spherically symmetric model atmospheres for giant stars with effective temperatures between 3000 and 6800~K

We present the extension of our NextGen model atmosphere grid to the regime of giant stars. The input physics of the models presented here is nearly identical to the NextGen dwarf atmosphere models, however spherical geometry is used self-consistently in the model calculations (including the radiative transfer). We re-visit the discussion of the effects of spherical geometry on the structure of the atmospheres and the emitted spectra and discuss the results of NLTE calculations for a few selected models.Comment: ApJ, in press (November 1999), 13 pages, also available at http://dilbert.physast.uga.edu/~yeti/PAPERS and at ftp://calvin.physast.uga.edu/pub/preprints/NG-giants.ps.g

Acute post-disaster medical needs of patients with diabetes: emergency department use in New York City by diabetic adults after Hurricane Sandy.

OBJECTIVE: To evaluate the acute impact of disasters on diabetic patients, we performed a geospatial analysis of emergency department (ED) use by New York City diabetic adults in the week after Hurricane Sandy. RESEARCH DESIGN AND METHODS: Using an all-payer claims database, we retrospectively analyzed the demographics, insurance status, and medical comorbidities of post-disaster ED patients with diabetes who lived in the most geographically vulnerable areas. We compared the patterns of ED use among diabetic adults in the first week after Hurricane Sandy\u27s landfall to utilization before the disaster in 2012. RESULTS: In the highest level evacuation zone in New York City, postdisaster increases in ED visits for a primary or secondary diagnosis of diabetes were attributable to a significantly higher proportion of Medicare patients. Emergency visits for a primary diagnosis of diabetes had an increased frequency of certain comorbidities, including hypertension, recent procedure, and chronic skin ulcers. Patients with a history of diabetes visited EDs in increased numbers after Hurricane Sandy for a primary diagnosis of myocardial infarction, prescription refills, drug dependence, dialysis, among other conditions. CONCLUSIONS: We found that diabetic adults aged 65 years and older are especially at risk for requiring postdisaster emergency care compared to other vulnerable populations. Our findings also suggest that there is a need to support diabetic adults particularly in the week after a disaster by ensuring access to medications, aftercare for patients who had a recent procedure, and optimize their cardiovascular health to reduce the risk of heart attacks