A (k,l)-colouring of a graph is a partition of its vertex set into k independent sets and l cliques. The bichromatic number chi^b of a graph is the minimum r$ such that the graph is (k,l)-colourable for all k+l=r. The bichromatic number is related to the cochromatic number, which can also be defined in terms of (k,l)-colourings. The bichromatic number is a fairly recent graph parameter that arises in the study of extremal graphs related to a classical result of Erd\H{o}s, Stone and Simonovits, and in the study of the edit distance of graphs from hereditary graph classes. While the cochromatic number has been well studied in the literature, there are only few known structural results for the bichromatic number. A main focus of this thesis is to establish a foundation of knowledge about the bichromatic number. The secondary focus is on $(k,l)$-colourings of certain interesting graph classes. Two known bounds for the bichromatic number are $\chi^b \leq \chi + \theta - 1$, where $\chi$ is the chromatic number and $\theta$ the clique covering number of the graph, and $\chi^b \geq \sqrt{n}$, where $n$ the number of vertices of the graph. We give a complete characterization of all graphs for which equality holds in the first bound, and show that the second bound is best possible by constructing graphs for square numbers $n$ such that equality holds in the bound. We investigate graphs for which the bichromatic number equals the cochromatic number and prove a Brooks-type theorem for the bichromatic number. Regarding $(k,l)$-colourings, we find a new algorithm for calculating the $(k,l)$-colourability of cographs and show that cographs have a particularly nice representation with regard to $(k,l)$-colourings. For proper circular arc graphs, we provide a method for $(k,l)$-colouring if $l \geq 1$, and establish an algebraic characterization for all maximally $(k,0)$-colourable proper circular arc graphs. Finally, we investigate the bichromatic number and cochromatic with respect to lexicographic products and show several nice bounds. / Graduate

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis, we define a certain group of order 160, which we call a hyperpentagonal group, and we prove that every totally real D5-extension of the rationals ramified at only one prime is contained in a hyperpentagonal extension of the rationals. This generalizes a result of Doud and Childers (originally conjectured by Wong) that every totally real S3 extension of the rationals ramified at only one prime is contained in an S4 extension.

Most heat transfer tubes are designed for either fully uniform wall temperature or fully uniform wall heat flux boundary conditions under forced convection. Several applications, including but not limited to the solar collectors of renewable energy systems, do however operate with non-uniform boundary conditions. Limited research has been conducted on non-uniform wall heat flux heat transfer coefficients in circular tubes, especially for mixed convection conditions. Such works are normally numerical in nature and little experimental work is available. In this experimental investigation the effects of the circumferential heat flux distribution and heat flux intensity on the single phase (liquid) internal heat transfer coefficient were considered for a horizontal circular tube. Focus was placed on the laminar flow regime of water within a stainless steel tube with an inner diameter of 27.8 mm and a length to diameter ratio of 72. Different outer wall heat flux conditions, including fully uniform and partially uniform heat fluxes were studied for Reynolds numbers ranging from 650 to 2 600 and a Prandtl number range of 4 to 7. The heat flux conditions included 360˚ (uniform) heating, lower 180˚ heating, upper 180˚ heating, 180˚ left and right hemispherical heating, lower 90˚ heating, upper 90˚ heating and slanted 180˚ heating. Depending on the angle span of the heating, local heat fluxes of 6 631 W/m2 , 4 421 W/m2 , 3 316 W/m2 , 2 210 W/m2 and 1 658 W/m2 were applied. Results indicate that the local and average steady state Nusselt numbers are greatly influenced by the applied heat flux position and intensity. Highest average heat transfer coefficients were achieved for case where the applied heat flux was positioned on the lower half (in terms of gravity) of the tubes circumference, while the lowest heat transfer coefficients were achieved when the heating was applied to the upper half of the tube. Variations in the heat transfer coefficient were found to be due to the secondary buoyancy induced flow effect. The relative thermal performance of the different heating scenarios where characterised and described by means of newly developed heat transfer coefficient correlations for fully uniform heating, lower 180° heating, and upper 180° heating. / Dissertation (MEng)--University of Pretoria, 2018. / Mechanical and Aeronautical Engineering / MEng / Unrestricted

Let (Formula presented.) be a graph. For two disjoint sets of vertices (Formula presented.) and (Formula presented.), set (Formula presented.) dominates set (Formula presented.) if every vertex in (Formula presented.) is adjacent to at least one vertex in (Formula presented.). In this paper we introduce the upper domatic number (Formula presented.), which equals the maximum order (Formula presented.) of a vertex partition (Formula presented.) such that for every (Formula presented.), (Formula presented.), either (Formula presented.) dominates (Formula presented.) or (Formula presented.) dominates (Formula presented.), or both. We study properties of the upper domatic number of a graph, determine bounds on (Formula presented.), and compare (Formula presented.) to a related parameter, the transitivity (Formula presented.) of (Formula presented.).

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let T be a tree of order n<2. We show that γt(T)≤a(T)+1, and we characterize the extremal trees achieving equality in this bound.

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let T be a tree of order n<2. We show that γt(T)≤a(T)+1, and we characterize the extremal trees achieving equality in this bound.

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. The total domination number yγ t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt (G) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. Haynes et al. (J. Combin. Math. Combin. Comput. 44 (2003) 115) showed that for any tree T of order at least 3, 1 ≤sdγt (T)≤3. In this paper, we give a constructive characterization of trees whose total domination subdivision number is 3.

Given a digraph D = (V, A), with vertex set V and arc set A, a set S ⊆ V is a dominating set if for every vertex v in V \ S, there are a vertex u in S and an arc (u, v) from u to v. In this chapter we consider the counterparts in directed graphs of independent, dominating, independent dominating, and total dominating sets in undirected graphs.

For a graph G=(V,E), we consider placing a variable number of pebbles on the vertices of V. A pebbling move consists of deleting two pebbles from a vertex u∈V and placing one pebble on a vertex v adjacent to u. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than some positive integer t pebbles and for any given vertex v∈V, it is possible, by a sequence of pebbling moves, to move at least one pebble to v. We relate this minimum number of pebbles to several other well-studied parameters of a graph G, including the domination number, the optimal pebbling number, and the Roman domination number of G.