Abstract For a set D of positive integers, the integral distance graph G(Z, D) is the graph with vertex set Z and edge set { xy : x, y 2 Z and | x − y | 2 D } . An integral distance graph G(Z, D) is called ¡§locally dense¡¨ if the clique size of G(Z, D) is not less to | D | . This paper acterizes locally dense integral distance graphs and determine their chromatic numbers.

This thesis discusses the game chromatic number of Halin graphs. We shall prove that with a few exceptions, all Halin graphs have game chromatic number 4.

In vivo monitoring of mouse models of colon cancer promises to reduce the cost of research by improving sacrifice timing and allowing serial studies that observe the progression of disease and drug efficacy in a relatively small set of animals. Optical coherence tomography (OCT) is an optical analog of ultrasound imaging, capable of minimally-invasive mapping of light scatter intensity up to 2 mm deep in tissue. In this work, factors limiting resolution in OCT were examined and devices were created and applied to mouse colon imaging that extended the state-of-the-art in endoscopic ultrahigh-resolution OCT. First, axial chromatic aberration of the objective optics acts as a spectral filter in the sample arm limiting the effective bandwidth of the system. An achromatized endoscope design was demonstrated that achieved axial resolution of 2.3 mum in tissue and 4.4 mum lateral spot diameter with 101 dB sensitivity when interfaced with a time domain OCT system utilizing a 10-femtosecond laser (bandwidth=150 nm FWHM, center wavelength=800 nm). Second, dispersion matching between the sample and reference arms presents the practical resolution limit to endoscopic implementations including a separate, fiber-based reference arm. A second endoscope incorporated the reference arm into the tip of the endoscope using a novel custom beamsplitter prism and achieved 2.4 mum axial resolution in tissue without adjustments for pathlength or dispersion matching when interfaced with a spectrometer-based frequency domain OCT system and a similar laser. Third, non-linear dispersion of the sample media with respect to wavelength causes distortion and broadening of the axial point spread function when data are sampled uniformly in optical frequency. An experiment was performed on high dispersion glass to demonstrate that dispersion artifact free imaging can be achieved without post process corrections if the samples are acquired at equal intervals of media index of refraction divided by vacuum wavelength. Finally, other microscopic modalities that depend on tissue scatter intensity are used to find the origins of scatter in the mouse colonic mucosa. These observations are used to explain unexpected features found in ultrahigh-resolution tomograms collected with the two endoscopes presented.

Reliable colour constancy by industry for colour conducted to quantify metamerism. and metamerism indices are highly desired quality control. Two experiments were the degree of colour constancy and In the colour constancy experiment, 240 wool samples were prepared and scaled using a magnitude estimation method by a panel of 5 experienced observers under sources D65, A and TL84. 2 corresponding data sets derived from the experimental results were used to test various chromatic adaptation transforms. The results clearly show that the BFD transform gave the most precise prediction than the other transforms. Attempts were also made to derive 4 new transforms from four independent data sets. These gave similar performance as that of the BFD, but overcome the BFO's problem (incapable of predicting some of the high saturated colours). Hence, these transforms should be used with confidence for predicting the degree of colour constancy. This experimental results were also used to test various uniform colour spaces and colour appearance models. The Hunt94 model gave the most precise prediction to the colourfulness and hue results. Modification was made to its lightness scale for improving the fit. In the metamerism experiment, 76 pairs of wool samples were prepared and assessed with 20 observations using a grey scale under 7 sources: D65, A, TL84, TL83, P27, W and WW. The experimental results were used to test 3 types of illuminant metamerism indices derived here. It was found that calculating colour difference using 3 colour difference formulae, i.e. CMC, BFD and CIE94 gave the most precise prediction to the visual results. The degree of precision is quite satisfactory in comparison with typical observer precision. A new standard deviate observer (SDO) was also derived. This together with the CIE SDO and 1964 Observer were tested using the author's and the Obande's data. The results showed that the new SDO predicted results more accurate than those from the other two CIE Observers. An Observer Metamerism Index (OMI) was also derived to indicate the degree of metamerism based upon the new SDO. The results showed that the new SDO was more suitable for indicating the degree of observer metamerism.

A colour may have a different appearance under different viewing conditions. This causes many problems in the colour reproduction industry. Thus the importance of prediction of colour appearance has arisen. In this study, a mathematical model to predict colour appearance was developed based on the investigation of the changes of colour appearance under a wide range of media and viewing conditions. The media studied included large cut-sheet transparency films, 35mm projected slides, reflection samples and monitor colours. The viewing conditions varied were light source, luminance level and viewing background. Colour appearance was studied using the magnitude estimation technique. In general, colours appeared more colourful, lighter and brighter with an increase in luminance level. Background and flare light had considerable influence on colour appearance for cut-sheet transparency media. Simultaneous contrast effects occurred when a monitor colour was displayed against a chromatic surround. The monitor colour appeared lighter with a darker induction field. When a coloured area was enlarged, lightness tended to increase while colourfulness tended to decrease. Colour appearance was also affected by the closest neighbouring colour. In this case, the hue of the colour largely shifted towards the direction of the opponent hue of the induction colour. The data obtained were applied to test three colour spaces and two colour appearance models. For reflection media, the Hunt91 model performed the best. However it was not satisfactory when applied to transmissive media. Based on these results, the Hunt93 model was developed by modification of the Hunt91 model. The new model widens the application range of the Hunt91 and is reversible.

Given an n-vertex graph Gand two positive integers d, k is an element of N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G(if one exists) with distinct numbers from 1 to kn(treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the (left perpendicular 2n/3 right pendicular, 2n)-differential coloring problem, even in the case where the input graph is planar.

No description available.

This thesis explores several of the major results in list-coloring in an expository fashion. As a specialization of list coloring, the sum list chromatic number is explored in detail. Ultimately, the thesis is designed to motivate the discussion of coloring problems and, hopefully, interest the reader in the branch of coloring problems in graph theory.

No / Physiological studies of colour vision have not yet resolved the controversial issue of how chromatic opponency is constructed at a neuronal level. Two competing theories, the cone-selective hypothesis and the random-wiring hypothesis, are currently equivocal to the architecture of the primate retina. In central vision, both schemes are capable of producing colour opponency due to the fact that receptive field centres receive input from a single bipolar cell ¿ the so called `private line arrangement¿. However, in peripheral vision this single-cone input to the receptive field centre is lost, so that any random cone connectivity would result in a predictable reduction in the quality of colour vision. Behavioural studies thus far have indeed suggested a selective loss of chromatic sensitivity in peripheral vision. We investigated chromatic sensitivity as a function of eccentricity for the cardinal chromatic (L/M and S/(L + M)) and achromatic (L + M) pathways, adopting stimulus size as the critical variable. Results show that performance can be equated across the visual field simply by a change of scale (size). In other words, there exists no qualitative loss of chromatic sensitivity across the visual field. Critically, however, the quantitative nature of size dependency for each of the cardinal chromatic and achromatic mechanisms is very specific, reinforcing their independence in terms of anatomy and genetics. Our data provide clear evidence for a physiological model of primate colour vision that retains chromatic quality in peripheral vision, thus supporting the cone-selective hypothesis.

This thesis discusses colouring, circular list colouring and adapted game colouring of graphs. For colouring, this thesis obtains a sufficient condition for a planar graph to be 3-colourable. Suppose G is a planar graph. Let H_G be the graph with vertex set V (H_G) = {C : C is a cycle of G with |C| ∈ {4, 6, 7}} and edge set E(H_G) = {CiCj : Ci and Cj have edges in common}. We prove that if any 3-cycles and 5-cycles are not adjacent to i-cycles for 3 ≤ i ≤ 7, and H_G is a forest, then G is 3-colourable. For circular consecutive choosability, this thesis obtains a basic relation among chcc(G), X(G) and Xc(G) for any finite graph G. We show that for any finite graph G, X(G) − 1 ≤ chcc(G) < 2 Xc(G). We also determine the value of chcc(G) for complete graphs, trees, cycles, balanced complete bipartite graphs and some complete multi-partite graphs. Upper and lower bounds for chcc(G) are given for some other classes of graphs. For adapted game chromatic number, this thesis studies the adapted game chromatic number of various classes of graphs. We prove that the maximum adapted game chromatic number of trees is 3; the maximum adapted game chromatic number of outerplanar graphs is 5; the maximum adapted game chromatic number of partial k-trees is between k + 2 and 2k + 1; and the maximum adapted game chromatic number of planar graphs is between 6 and 11. We also give upper bounds for the Cartesian product of special classes of graphs, such as the Cartesian product of partial k-trees and outerplanar graphs, or planar graphs.