Thesis (M.Sc.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2009/09/07). Includes bibliographical references.

This dissertation is in two parts, each of three chapters. In Part 1, I shall prove some results concerning variants of the 'secretary problem'. In Part 2, I shall bound several generalizations of the acyclic chromatic number of a graph as functions of its maximum degree. I shall begin Chapter 1 by describing the classical secretary problem, in which the aim is to select the best candidate for the post of a secretary, and its solution. I shall then summarize some of its many generalizations that have been studied up to now, provide some basic theory, and briefly outline the results that I shall prove. In Chapter 2, I shall suppose that the candidates come as ‘m’ pairs of equally qualified identical twins. I shall describe an optimal strategy, a formula for its probability of success and the asymptotic behaviour of this strategy and its probability of success as m → ∞. I shall also find an optimal strategy and its probability of success for the analagous version with ‘c’-tuplets. I shall move away from known posets in Chapter 3, assuming instead that the candidates come from a poset about which the only information known is its size and number of maximal elements. I shall show that, given this information, there is an algorithm that is successful with probability at least ¹/e . For posets with ‘k ≥ 2’ maximal elements, I shall prove that if their width is also ‘k’ then this can be improved to ‘k-1√1/k’ and show that no better bound of this type is possible. In Chapter 4, I shall describe the history of acyclic colourings, in which a graph must be properly coloured with no two-coloured cycle, and state some results known about them and their variants. In particular, I shall highlight a result of Alon, McDiarmid and Reed, which bounds the acyclic chromatic number of a graph by a function of its maximum degree. My results in the next two chapters are of this form. I shall consider two natural generalizations in Chapter 5. In the first, only cycles of length at least ’l’ must receive at least three colours. In the second, every cycle must receive at least ‘c’ colours, except those of length less than ‘c’, which must be multicoloured. My results in Chapter 6 generalize the concept of a cycle; it is now subgraphs with minimum degree ‘r’ that must receive at least three colours, rather than subgraphs with minimum degree two (which contain cycles). I shall also consider a natural version of this problem for hypergraphs.

In Chapter 1 the classical secretary problem is introduced. Chapters 2 and 3 are variations of this problem. Chapter 2, discusses the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter p and finite time horizon n. The optimal strategy (continue or stop) depends on a sequence of threshold values (critical probabilities) which has an oscillating pattern. Several properties of this sequence have been proved by Dr. Allaart. Further properties have been recently proved. In Chapter 3, a gambler will observe a finite sequence of continuous random variables. After he observes a value he must decide to stop or continue taking observations. He can play two different games A) Win at the maximum or B) Win within a proportion of the maximum. In the first section the sequence to be observed is independent. It is shown that for each n>1, theoptimal win probability in game A is bounded below by (1-1/n)^{n-1}. It is accomplished by reducing the problem to that of choosing the maximum of a special sequence of two-valued random variables and applying the sum-the-odds theorem of Bruss (2000). Secondly, it is assumed the sequence is i.i.d. The best lower bounds are provided for the winning probabilities in game B given any continuous distribution. These bounds are the optimal win probabilities of a game A which was examined by Gilbert and Mosteller (1966).

Robbins' problem is an optimal stopping problem where one seeks to minimize the expected rank of their observations among all observations. We examine random walk analogs to Robbins' problem in both discrete and continuous time. In discrete time, we consider full information and relative ranks versions of this problem. For three step walks, we give the optimal stopping rule and the expected rank for both versions. We also give asymptotic upper bounds for the expected rank in discrete time. Finally, we give upper and lower bounds for the expected rank in continuous time, and we show that the expected rank in the continuous time problem is at least as large as the normalized asymptotic expected rank in the full information discrete time version.