Date of Award
Doctor of Philosophy
Paul Neville Balister
Ralph Jasper Faudree
Anna Helena Kamińska
This dissertation consists of three results from the domain of extremal combinatorics. While they are all interconnected, we can distinguish two major areas: separating families and combinatorial games. In the following we shall outline them in further detail. Searching is a fundamental tasks inherent to human life both past and present. With the rise of computers and information technology, it was necessary to put all these related tasks into a unified mathematical framework. It fell to Rényi and Katona in the 1960s to pioneer this field. Providing what is nowadays common knowledge about binary search trees, and standard programming approaches in load balancing like ‘divide and conquer’, Katona had the foresight (then mostly motivated by medical examples) to explore parallelised search. His celebrated result not only provided bounds for searching for a unique unknown element, it also provided a specific way construct such a search algorithm. The proof relied heavily on analytic methods and optimisation. Surprisingly it would take more than 40 years for Hosszu, Tapolcai, and Wiener, to simplify the proof using purely matrix theoretic methods. In Chapter 1 we generalise their work to the search for unknown sets of general size. While our principal approach is similar, we need to embed this problem into a framework of regular graphs and hypergraphs with minimal girth conditions. We now turn to the second part of this dissertation, where we will consider two variants of combinatorial games. Chapter 2 is concerned with the 1-person game of Flood-It. Given a vertex-coloured graph, we seek to obtain a monochromatic graph by recolouring its monochromatic components. Having started out as an internet game, it became of interest in terms of computational complexity and the previous work addresses that, confirming for most instances of this game NP-completeness. We take a different approach, more in line with extremal combinatorics. We investigate how many moves are necessary to achieve the goal and establish two kinds of upper bounds for that question. We show that they are tight for a surprisingly vast array of graph classes, yet for blow-ups of paths and cycles the time necessary is the same as for the original graph. Furthermore, we investigate square grids, providing a partial solution.Related to both previous chapters, Chapter 3 turns to a combinatorial game, called Cops and Robber, a pursuit and evasion game on a finite graph. Over the years has found its place in the classical canon of combinatorial problems, mainly due to a longstanding open conjecture due to Meyniel. As such, it is related to both previous chapters. We are interested in establishing the cop number and the capture time for certain classes of graphs. In this chapter, we extend previous results for grids to the torus and give an explicit capturing algorithm. Chapter 1 is based on joint work with F.S. Benevides, D. Gerbner, and C.T. Palmer, Chapter 2 was investigated jointly with K. Meeks and Chapter 3 was researched together with S. Koch.
dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.
Vu, Dominik Kim, "Separating Families and Combinatorial Games" (2014). Electronic Theses and Dissertations. 1096.