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A range of nitrosylmetalloporphyrin complexes, M(porph)NO [M = Fe, Co; porph = TPP, OEP, T[PIV]PP], some labelled with [15]NO, has been synthesised and investigated as structural and spectroscopic models of the cured meat pigment, nitrosylmyoglobin. The nitrosylmetalloporphyrins were found to be unstable in solution (and some in the solid state) with respect to loss or further reaction of the nitrosyl ligand. The diamagnetic cobalt (III) porphyrin nitrosyIs and a range of other square pyramidal complexes of cobalt(III) with an apical nitrosyl ligand have been further investigated by [15]N and [59]Co NMR spectroscopy to establish whether the CoNO group is bent or linear and to study nephelauxetic and spectrochemical effects at the metal. In all cases studied the CoNO group is bent (CoNO ~ 120°). The basal ligands were dithiocarbamate, tetradentate Schiff base, porphyrin, and bidentate diamine and oximate giving S[4], S[2]N[2], N[4], OONN, or ONON coordination in the equatorial plane. The shielding of both cobalt and nitrogen tends to decrease with decrease in M(d) to pi* (NO) back-bonding as indicated by M-N and N-O bond distances, and the MNO angle and the NO stretching frequencies. The shieldings decrease with electron withdrawal by ring substituents and also in going from sulphur to nitrogen to oxygen coligands, i.e. with decrease in ligand field splitting and in the nephelauxetism of the coligands. The bent nitrosyls of cobalt can function as oxygen transfer catalysts via a nitrosyl-nitro couple. Consequently, a range of Co(III) porphyrin nitro complexes, (L)Co(porph)NO[2] (porph = TPP, OEP; L = py, 1Melm) and several nitro complexes containing the other basal ligands have been synthesised and investigated by the [15]N and [59]Co NMR technique. The coordinated nitro ligand is labile and linkage isomerism and ligand exchange have been observed. These observations may be related to the deterioration of cured meat colour which is of importance to the food industry. The crystal structure of (py)Co(TPP)NO[2] is presented; the nitro group of which is rotationally disordered. Recrystallisation of (1MeIm)Co-(TPP)NO[2] gave [Co(TPP)(1Melm)[2]]NO[2], the crystal structure of which has been determined. Formation of this nitrite is further evidence for the lability of the nitro ligand. The known complex Fe(salphen)NO was prepared using ascorbic acid for the first time to prevent contamination with the oxo-bridged iron(III) dimer. Variable temperature [57]Fe Mossbauer effect studies of the S = 1/2 to S = 3/2 spin transition for Fe(salphen)NO show that the crossover is continuous and centred at T[c] ~ 181 K, and that the electronic transition between the two spin states is rapid on the Mossbauer timescale. The fluxionality of CoCl[2]([15]NO)(PPh[2]Me)[2] was studied by variable temperature [15]N and [31]P NMR. An [17]O NMR study of the ?-peroxo complex [(NH[3])[5]Co[17]O[2]Co(NH[3])[5]](NO[3])[4]. 2H[2]O is presented. An attempt to measure [15]N NMR spectra of the diamagnetic 18 electron species, Fe(porph)([14]NO)([15]NO) (porph = TPP, OEP) presumably containing linear FeNO[+] and bent FeNO[-] was unsuccessful.

This thesis was originally motivated by considering vector space analogues of problems in extremal set theory, but our main results concern colouring a graph that is intimately related to these vector space analogues. The vertices of the <em>q</em>-Kneser graph are the <em>k</em>-dimensional subspaces of a vector space of dimension <em>v</em> over F_<em>q</em>, and two <em>k</em>-subspaces are adjacent if they have trivial intersection. The new results in this thesis involve colouring the <em>q</em>-Kneser graph when <em>k</em>=2. There are two cases. When <em>k</em>=2 and <em>v</em>=4, the chromatic number is <em>q</em>²+<em>q</em>. If <em>k</em>=2 and <em>v</em>>4, the chromatic number is (<em>q</em>^(v-1)-1)/(<em>q</em>-1). In both cases, we characterise the minimal colourings. We develop some theory for colouring the <em>q</em>-Kneser graph in general.

This thesis was originally motivated by considering vector space analogues of problems in extremal set theory, but our main results concern colouring a graph that is intimately related to these vector space analogues. The vertices of the <em>q</em>-Kneser graph are the <em>k</em>-dimensional subspaces of a vector space of dimension <em>v</em> over F_<em>q</em>, and two <em>k</em>-subspaces are adjacent if they have trivial intersection. The new results in this thesis involve colouring the <em>q</em>-Kneser graph when <em>k</em>=2. There are two cases. When <em>k</em>=2 and <em>v</em>=4, the chromatic number is <em>q</em>²+<em>q</em>. If <em>k</em>=2 and <em>v</em>>4, the chromatic number is (<em>q</em>^(v-1)-1)/(<em>q</em>-1). In both cases, we characterise the minimal colourings. We develop some theory for colouring the <em>q</em>-Kneser graph in general.

We consider a generalisation of proper vertex colouring of graphs, referred to as improper colouring, in which each vertex can only be adjacent to a bounded number t of vertices with the same colour, and we study this type of graph colouring problem in several different settings. The thesis is divided into six chapters. In Chapter 1, we outline previous work in the area of improper colouring. In Chapters 2 and 3, we consider improper colouring of unit disk graphs -- a topic motivated by applications in telecommunications -- and take two approaches, first an algorithmic one and then an average-case analysis. In Chapter 4, we study the asymptotic behaviour of the improper chromatic number for the classical Erdos-Renyi model of random graphs. In Chapter 5, we discuss acyclic improper colourings, a specialisation of improper colouring, for graphs of bounded maximum degree. Finally, in Chapter 6, we consider another type of colouring, frugal colouring, in which no colour appears more than a bounded number of times in any neighbourhood. Throughout the thesis, we will observe a gradient of behaviours: for random unit disk graphs and "large" unit disk graphs, we can greatly reduce the required number of colours relative to proper colouring; in Erdos-Renyi random graphs, we do gain some improvement but only when t is relatively large; for acyclic improper chromatic numbers of bounded degree graphs, we discern an asymptotic difference in only a very narrow range of choices for t.

Image based scene reconstruction from multiple views is an interesting challenge, with many ambiguities and sources of noise. One approach to scene reconstruction is Voxel Colouring, Seitz and Dyer [26], which uses colour information in images and handles the problem of occlusion. Culbertson and Malzbender [11], introduced Generalised Voxel Colouring (GVC) which uses projection and rasterization to establish global scene visibility. Our work has involved investigating the use of ray traversal as an efficient alternative. We have developed two main approaches along this line, Ray Images and Ray Buckets. Comparisons between implementations of our algorithms and variations of GVC are presented, as well as applications to areas of optimisation colour consistency and level of detail. Ray traversal seems a promising approach to scene visibility, but requires more work to be of practical use. Our methods show some advantages over existing approaches in time use. However we have not been as succesful as an- ticipated in reconstruction quality shown by implementation of optimisation colour consistency.

The aim of this thesis was to analyze the genetic foundation of color in a selected population of horses in Czech Republic. For the analysis, I chose horse breed Paint horse. I characterized this breed, dealing with pigmentation and characteristic colors and patterns in the Paint horse. I also described the methods that I then used in the practical part. It was collected 11 blood samples from horses, in which was detected the presence of KIT gene mutation.