The diploma thesis deals with the influence of colouring additives and setting of the process parameters of 3D printing on the mechanical and surface properties of samples made of PLA material. The work describes the process of filament production, as well as the printing of normalized samples on a 3D printer using the additive method Fused Deposition Modeling. The impact of 3 types of colouring additives is evaluated on the basis of tensile test, hardness test and surface analysis. The evaluated quantities are primarily tensile strength, hardness, surface texture, roughness and corrugation. The work also evaluates the influence of the percentage of sample filling, the direction of the fibres of the inner filling and the orientation of the samples in the printing chamber of the 3D printer on the mechanical properties. The results are evaluated on the basis of the tensile test and the evaluated quantities are mainly the tensile strength, the ultimate stress and the modulus of elasticity in traction. The work is completed by evaluating the results and overall recommendations for filament manufacturers and users.

Cette thèse a pour thème la coloration de diverses classes de graphes épars. Shearer montra en 1983 [She83] que le ratio d'indépendance des graphes sans triangle de degré maximal d est au moins (1-o(1))ln d/d, et 13 ans plus tard Johansson [Joh96] démontra que le nombre chromatique de ces graphes est au plus O(d/ln d) quand d tend vers l'infini. Ce dernier résultat fut récemment amélioré par Molloy [Mol19], qui montra que la borne (1+o(1))d/ln d est valide quand d tend vers l'infini.Tandis que le résultat de Molloy s'exprime à l'aide d'un paramètre global, le degré maximal du graphe, nous montrons qu'il est possible de l'étendre à la coloration locale. Il s'agit de la coloration par liste, où la taille de la liste associée à chaque sommet ne dépend que de son degré. Avec une méthode différente se basant sur les propriétés de la distribution hard-core sur les ensembles indépendants d'un graphe, nous obtenons un résultat similaire pour la coloration fractionnaire locale, avec des hypothèses plus faibles. Nous démontrons également un résultat concernant la coloration fractionnaire locale des graphes où chaque sommet est contenu dans un nombre borné de triangles, et une borne principalement optimale sur le taux d'occupation — la taille moyenne des ensembles indépendants — de ces graphes. Nous considérons également les graphes de maille 7, et prouvons des résultats similaires qui améliorent les bornes précédemment connues quand le degré maximal du graphe est au plus 10^7. Finalement, pour les graphes d-réguliers où d vaut 3, 4, ou 5, de maille g variant entre 6 et 12, nous démontrons de nouvelles bornes inférieures sur le ratio d'indépendance.Le Chapitre 2 est dédié à la coloration à distance t d'un graphe, qui généralise la notion de coloration forte des arêtes. Nous cherchons à étendre le théorème de Johansson à la coloration à distance t, par l'exclusion de certains cycles. Le résultat de Johansson s'obtient par exclusion des triangles, ou des cycles de taille k pour n'importe quelle valeur de k. Nous montrons que l'exclusion des cycles de taille 2k, pour n'importe quel k>t, a un effet similaire sur le nombre chromatique à distance t, et sur l'indice chromatique à distance t+1. En outre, quand t est impair, une conclusion similaire peut se faire pour le nombre chromatique à distance t par l'exclusion des cycles de d'une taille impaire fixée valant au moins 3t. Nous étudions l'optimalité de ces résultats à l'aide de constructions de nature combinatoire, algébrique, et probabiliste.Dans le Chapitre 3, nous nous intéressons à la densité bipartie induite des graphes sans triangle, un paramètre relaxant celui de la coloration fractionnaire. Motivés par une conjecture de Esperet, Kang, et Thomassé [EKT19], qui prétend que la densité bipartie induite de graphes sans triangle de degré moyen d est au moins de l'ordre de ln d, nous démontrons cette conjecture quand d est suffisamment grand en termes du nombre de sommets n, à savoir d est au moins de l'ordre de (n ln n)^(1/2). Ce résultat ne pourrait être amélioré que par une valeur de l'ordre de ln n, ce que nous montrons à l'aide d'une construction reposant sur le processus sans triangle. Nos travaux se ramènent à un problème intéressant, celui de déterminer le nombre chromatique fractionnaire maximal d'un graphe épars à n sommets. Nous prouvons des bornes supérieures non triviales pour les graphes sans triangle, et pour les graphes dont chaque sommet appartient à un nombre borné de triangles.Cette thèse est reliée aux nombres de Ramsey. À ce jour, le meilleur encadrement connu sur R(3,t) nous est donné par le résultat de Shearer, et par une analyse récente du processus sans triangle [BoKe13+,FGM13+], ce qui donne(1-o(1)) t²/(4 ln t) < R(3,t) < (1+o(1)) t²/ln t. (1)Beaucoup de nos résultats ne pourraient être améliorés à moins d'améliorer par la même occasion (1), ce qui constituerait une révolution dans la théorie de Ramsey quantitative. / This thesis focuses on generalisations of the colouring problem in various classes of sparse graphs.Triangle-free graphs of maximum degree d are known to have independence ratio at least (1-o(1))ln d/d by a result of Shearer [She83], and chromatic number at most O(d/ln d) by a result of Johansson [Joh96], as d grows to infinity. This was recently improved by Molloy, who showed that the chromatic number of triangle-free graphs of maximum degree d is at most (1+o(1))d/ln d as d grows to infinity.While Molloy's result is expressed with a global parameter, the maximum degree of the graph, we first show that it is possible to extend it to local colourings. Those are list colourings where the size of the list associated to a given vertex depends only on the degree of that vertex. With a different method relying on the properties of the hard-core distribution on the independent sets of a graph, we obtain a similar result for local fractional colourings, with weaker assumptions. We also provide an analogous result concerning local fractional colourings of graphs where each vertex is contained in a bounded number of triangles, and a sharp bound for the occupancy fraction — the average size of an independent set — of those graphs. In another direction, we also consider graphs of girth 7, and prove related results which improve on the previously known bounds when the maximum degree does not exceed 10^7. Finally, for d-regular graphs with d in the set {3,4,5}, of girth g varying between 6 and 12, we provide new lower bounds on the independence ratio.The second chapter is dedicated to distance colourings of graphs, a generalisation of strong edge-colourings. Extending the theme of the first chapter, we investigate minimal sparsity conditions in order to obtain Johansson-like results for distance colourings. While Johansson's result follows from the exclusion of triangles — or actually of cycles of any fixed length — we show that excluding cycles of length 2k, provided that k>t, has a similar effect for the distance-t chromatic number and the distance-(t+1) chromatic index. When t is odd, the same holds for the distance-t chromatic number by excluding cycles of fixed odd length at least 3t. We investigate the asymptotic sharpness of our results with constructions of combinatorial, algebraic, and probabilistic natures.In the third chapter, we are interested in the bipartite induced density of triangle-free graphs, a parameter which conceptually lies between the independence ratio and the fractional chromatic number. Motivated by a conjecture of Esperet, Kang, and Thomassé [EKT19], which states that the bipartite induced density of a triangle-free graph of average degree d should be at least of the order of ln d, we prove that the conjecture holds for when d is large enough in terms of the number of vertices n, namely d is at least of the order of (n ln n)^(1/2). Our result is shown to be sharp up to term of the order of ln n, with a construction relying on the triangle-free process. Our work on the bipartite induced density raises an interesting related problem, which aims at determining the maximum possible fractional chromatic number of sparse graph where the only known parameter is the number of vertices. We prove non trivial upper bounds for triangle-free graphs, and graphs where each vertex belongs to a bounded number of triangles.All the content of this thesis is a collection of specialisations of the off-diagonal Ramsey theory. To this date, the best-known bounds on the off-diagonal Ramsey number R(3,t) come from the aforementioned result of Shearer for the upper-bound, and a recent analysis of the triangle-free process [BoKe13+,FGM13+] for the lower bound, giving(1-o(1)) t²/(4 ln t) < R(3,t) < (1+o(1)) t²/ln t. (1)Many of our results are best possible barring an improvement of (1), which would be a breakthrough in off-diagonal Ramsey theory.

Within Alinea, the purpose is to explore the use of bioplastic as unconventional yarn in the traditional technique of handweaving. The focus toward bioplastic as a design material and the technique of handweaving as the fabrication technique to generate broader alternatives for using bioplastic materials in woven textile design. Described through experimental and practise-based research, handwoven bioplastic samples have been explored to investigate the methods of structures and bindings, gradient colouring and print design within bioplastic and weaving. With the attempt to make bioplastic more accessible for the textile industry. The experimental design research resulted in scaled prototypes that showcase a collection of seven pieces that present various design possibilities and potentials regarding bioplastic within the textile weaving technique, including distinct structural tactile qualities bioplastics can offer to the field of textile. It can be concluded that bioplastic can play a role in becomes a desirable material steering textile design towards a more sustainable future in the textile design field. Furthermore, give handwoven materials new aesthetics by producing unique structures and tactile features.

Graph colourings and combinatorial games are two very widely studied topics in discrete mathematics. This thesis addresses the computational complexity of a range of problems falling within one or both of these subjects. Much of the thesis is concerned with the computational complexity of problems related to the combinatorial game (Free-)Flood-It, in which players aim to make a coloured graph monochromatic ("flood" the graph) with the minimum possible number of flooding operations; such problems are known to be computationally hard in many cases. We begin by proving some general structural results about the behaviour of the game, including a powerful characterisation of the number of moves required to flood a graph in terms of the number of moves required to flood its spanning trees; these structural results are then applied to prove tractability results about a number of flood-filling problems. We also consider the computational complexity of flood-filling problems when the game is played on a rectangular grid of fixed height (focussing in particular on 3xn and 2xn grids), answering an open question of Clifford, Jalsenius, Montanaro and Sach. The final chapter concerns the parameterised complexity of list problems on graphs of bounded treewidth. We prove structural results determining the list edge chromatic number and list total chromatic number of graphs with bounded treewidth and large maximum degree, which are special cases of the List (Edge) Colouring Conjecture and Total Colouring Conjecture respectively. Using these results, we show that the problem of determining either of these quantities is fixed parameter tractable, parameterised by the treewidth of the input graph. Finally, we analyse a list version of the Hamilton Path problem, and prove it to be W[1]-hard when parameterised by the pathwidth of the input graph. These results answer two open questions of Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen.

We study graph parameters arising from different types of colourings of random graphs, defined broadly as an assignment of colours to either the vertices or the edges of a graph. The chromatic number X(G) of a graph is the minimum number of colours required for a vertex colouring where no two adjacent vertices are coloured the same. Determining the chromatic number is one of the classic challenges in random graph theory. In Chapter 3, we give new upper and lower bounds for the chromatic number of the dense random graph G(n,p)) where p &isin; (0,1) is constant. These bounds are the first to match up to an additive term of order o(1) in the denominator, and in particular, they determine the average colour class size in an optimal colouring up to an additive term of order o(1). In Chapter 4, we study a related graph parameter called the equitable chromatic number. This is defined as the minimum number of colours needed for a vertex colouring where no two adjacent vertices are coloured the same and, additionally, all colour classes are as equal in size as possible. We prove one point concentration of the equitable chromatic number of the dense random graph G(n,m) with m = pn(n-1)/2, p &LT; 1-1/e² constant, on a subsequence of the integers. We also show that whp, the dense random graph G(n,p) allows an almost equitable colouring with a near optimal number of colours. We call an edge colouring of a graph G a rainbow colouring if every pair of vertices is joined by a rainbow path, which is a path where no colour is repeated. The least number of colours where this is possible is called the rainbow connection number rc(G). Since its introduction in 2008 as a new way to quantify how well connected a given graph is, the rainbow connection number has attracted the attention of a great number of researchers. For any graph G, rc(G)&ge;diam(G), where diam(G) denotes the diameter. In Chapter 5, we will see that in the random graph G(n,p), rainbow connection number 2 is essentially equivalent to diameter 2. More specifically, we consider G ~ G(n,p) close to the diameter 2 threshold and show that whp rc(G) = diam(G) &isin; {2,3}. Furthermore, we show that in the random graph process, whp the hitting times of diameter 2 and of rainbow connection number 2 coincide. In Chapter 6, we investigate sharp thresholds for the property rc(G)&le;=r where r is a fixed integer. The results of Chapter 6 imply that for r=2, the properties rc(G)&le;=2 and diam(G)&le;=2 share the same sharp threshold. For r&ge;3, the situation seems quite different. We propose an alternative threshold and prove that this is an upper bound for the sharp threshold for rc(G)&le;=r where r&ge;3.

Thesis (M. Sc. (Horticulture)) -- University of Limpopo, 2018 / ‘Hass’ avocado fruit changes skin colour from green to purple and then black during ripening. However, markets importing South African avocado fruit have been complaining about the ‘Hass’ skin colour not changing to purple/black during ripening. Thus, the study aimed to investigate the effect of tree girdling, harvest time and ripening temperature on ‘Hass’ avocado fruit skin colour development during ripening. The mature ‘Hass’ avocado fruit were harvested from girdled and non-girdled trees during early (April), mid- (May) and late (June) harvest times. Upon arrival, in the laboratory fruit were cold stored at 5.5°C for 28 days. After storage, fruit were ripened at 25, 21 and 16°C for 8, 6 and 4 days, respectively. After withdrawal from clod storage fruit were evaluated for skin colour development, ripening and physiological disorders (chilling injury). Fruit from girdled trees showed high maturity (low moisture content) when compared with fruit from non-girdled trees during early and mid-harvest. With respect to skin colour development, the results indicate that skin eye colour development of fruit from girdled and non-girdled trees minimally increased from emerald green (1) to olive green (3) across all harvest times, ripening temperature and ripening duration. However, late season fruit from non-girdled trees improved to purple (4) when ripened at 21°C when compared with fruit from girdled trees. In terms of objective colour, lightness, hue angle and chroma decreased for fruit from girdled and non-girdled trees, across all harvest times, ripening temperature and ripening duration. Lightness and hue angle of fruit from girdled trees were slightly reduced when compared with fruit from non-girdled trees, throughout all harvest times, ripening temperature and duration. Early and mid-season fruit harvested from girdled trees showed rapid decrease of chroma when compared with fruit from non-girdled trees, throughout ripening temperature and x duration. In terms of softening, fruit from girdled trees showed higher firmness loss and ripening percentage within 6 (16°C) and 4 (21 and 25°C) days when compared with fruit from non-girdled trees during early and mid-harvest, whereas, late harvest fruit from girdled trees reached higher ripening percentage and firmness loss within 4 days throughout ripening temperatures. With respect to cold damage, late harvested fruit from girdled trees showed higher external chilling injury when compared with non-girdled trees, throughout ripening temperature. In general, girdling treatment improved fruit maturity, ripening rate and firmness loss. However, the incidence of variable skin colouring of ‘Hass’ avocado fruit during ripening was also prevalent in early harvested fruit from girdled tree, irrespective of ripening temperature. Keywords: girdling, harvest time, physiological disorder, ripening temperature, variable colouring

Thesis (M.Sc. (Horticulture)) --University of Limpopo, 2018 / ‘Hass’ avocado fruit changes skin colour from green to purple and then black during ripening. However, markets importing South African avocado fruit have been complaining about the ‘Hass’ skin colour not changing to purple/black during ripening. Thus, the study aimed to investigate the effect of tree girdling, harvest time and ripening temperature on ‘Hass’ avocado fruit skin colour development during ripening. The mature ‘Hass’ avocado fruit were harvested from girdled and non-girdled trees during early (April), mid- (May) and late (June) harvest times. Upon arrival, in the laboratory fruit were cold stored at 5.5°C for 28 days. After storage, fruit were ripened at 25, 21 and 16°C for 8, 6 and 4 days, respectively. After withdrawal from clod storage fruit were evaluated for skin colour development, ripening and physiological disorders (chilling injury). Fruit from girdled trees showed high maturity (low moisture content) when compared with fruit from non-girdled trees during early and mid-harvest. With respect to skin colour development, the results indicate that skin eye colour development of fruit from girdled and non-girdled trees minimally increased from emerald green (1) to olive green (3) across all harvest times, ripening temperature and ripening duration. However, late season fruit from non-girdled trees improved to purple (4) when ripened at 21°C when compared with fruit from girdled trees. In terms of objective colour, lightness, hue angle and chroma decreased for fruit from girdled and non-girdled trees, across all harvest times, ripening temperature and ripening duration. Lightness and hue angle of fruit from girdled trees were slightly reduced when compared with fruit from non-girdled trees, throughout all harvest times, ripening temperature and duration. Early and mid-season fruit harvested from girdled trees showed rapid decrease of chroma when compared with fruit from non-girdled trees, throughout ripening temperature and x duration. In terms of softening, fruit from girdled trees showed higher firmness loss and ripening percentage within 6 (16°C) and 4 (21 and 25°C) days when compared with fruit from non-girdled trees during early and mid-harvest, whereas, late harvest fruit from girdled trees reached higher ripening percentage and firmness loss within 4 days throughout ripening temperatures. With respect to cold damage, late harvested fruit from girdled trees showed higher external chilling injury when compared with non-girdled trees, throughout ripening temperature. In general, girdling treatment improved fruit maturity, ripening rate and firmness loss. However, the incidence of variable skin colouring of ‘Hass’ avocado fruit during ripening was also prevalent in early harvested fruit from girdled tree, irrespective of ripening temperature. Keywords: girdling, harvest time, physiological disorder, ripening temperature, variable colouring

Given a set of unit disks in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the disks? Richard Rado conjectured 1/4 and proved 1/4.41. In this thesis, we consider a variant of this problem where the disjointness constraint is relaxed: selected disks must be k-colourable with disks of the same colour pairwise-disjoint. Rado's problem is then the case where k = 1, and we focus our investigations on what can be proven for k > 1. Motivated by the problem of channel-assignment for Wi-Fi wireless access points, in which the use of 3 or fewer channels is a standard practice, we show that for k = 3 we can cover at least 1/2.09 and for k = 2 we can cover at least 1/2.82. We present a randomized algorithm to select and colour a subset of n disks to achieve these bounds in O(n) expected time. To achieve the weaker bounds of 1/2.77 for k = 3 and 1/3.37 for k = 2 we present a deterministic O(n^2) time algorithm. We also look at what bounds can be proven for arbitrary k, presenting two different methods of deriving bounds for any given k and comparing their performance. One of our methods is an extension of the method used to prove bounds for k = 2 and k = 3 above, while the other method takes a novel approach. Rado's proof is constructive, and uses a regular lattice positioned over the given set of disks to guide disk selection. Our proofs are also constructive and extend this idea: we use a k-coloured regular lattice to guide both disk selection and colouring. The complexity of implementing many of the constructions used in our proofs is dominated by a lattice positioning step. As such, we discuss the algorithmic issues involved in positioning lattices as required by each of our proofs. In particular, we show that a required lattice positioning step used in the deterministic O(n^2) algorithm mentioned above is 3SUM-hard, providing evidence that this algorithm is optimal among algorithms employing such a lattice positioning approach. We also present evidence that a similar lattice positioning step used in the constructions for our better bounds for k = 2 and k = 3 may not have an efficient exact implementation.

Given a set of unit disks in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the disks? Richard Rado conjectured 1/4 and proved 1/4.41. In this thesis, we consider a variant of this problem where the disjointness constraint is relaxed: selected disks must be k-colourable with disks of the same colour pairwise-disjoint. Rado's problem is then the case where k = 1, and we focus our investigations on what can be proven for k > 1. Motivated by the problem of channel-assignment for Wi-Fi wireless access points, in which the use of 3 or fewer channels is a standard practice, we show that for k = 3 we can cover at least 1/2.09 and for k = 2 we can cover at least 1/2.82. We present a randomized algorithm to select and colour a subset of n disks to achieve these bounds in O(n) expected time. To achieve the weaker bounds of 1/2.77 for k = 3 and 1/3.37 for k = 2 we present a deterministic O(n^2) time algorithm. We also look at what bounds can be proven for arbitrary k, presenting two different methods of deriving bounds for any given k and comparing their performance. One of our methods is an extension of the method used to prove bounds for k = 2 and k = 3 above, while the other method takes a novel approach. Rado's proof is constructive, and uses a regular lattice positioned over the given set of disks to guide disk selection. Our proofs are also constructive and extend this idea: we use a k-coloured regular lattice to guide both disk selection and colouring. The complexity of implementing many of the constructions used in our proofs is dominated by a lattice positioning step. As such, we discuss the algorithmic issues involved in positioning lattices as required by each of our proofs. In particular, we show that a required lattice positioning step used in the deterministic O(n^2) algorithm mentioned above is 3SUM-hard, providing evidence that this algorithm is optimal among algorithms employing such a lattice positioning approach. We also present evidence that a similar lattice positioning step used in the constructions for our better bounds for k = 2 and k = 3 may not have an efficient exact implementation.

No description available.