Date of Award
Doctor of Philosophy
This dissertation concerns several problems in graph theory, three of which involve grids and grid-like structures. The first is a variant of the game of cops and robbers, originally introduced independently in 1983 by Nowakowski and Winkler and by Quillot. We focus on a variant on the n×n grid in which the robber may move faster than the cops, providing upper and lower bounds on the cop number, the minimum number of cops for which the cops have a winning strategy in this perfect information pursuit-evasion game. The second problem concerns 2-neighbor bootstrap percolation on the n×n grid, denoted [n]2. Introduced in 1979 by Chalupa, Leith, and Reich, the process has been the subject of much study in both the probabilistic and deterministic models, and in particular on the grid. Recently, several extremal results have been proven regarding the maximum percolation time on [n]2, and here we provide upper and lower bounds on the minimum percolation time. The third problem concerns the relationship between the radius and the minimum degree of a graph. In 1987, Erdos, Pach, Pollock, and Tuza gave bounds on the diameter and radius of a graph, given its order and minimum degree. They also proved analagous results regarding triangle-free and C4-free graphs. Recently, Kim, Rho, Song, and Hwang improved their general bound. In this dissertation, we improve the bound in the triangle-free case. The last problem concerns Latin fillings of tableaux. The wide partition conjecture originated in an attempt to generalize Rota's basis conjecture, but even the weakest version, involving free matroids, remains open. Focusing on the connections between colorings and orientations of graphs demonstrated by Alon and Tarsi, we show that the line graph of a tableau's corresponding bipartite graph may be oriented so that it satisfies a certain desirable degree condition.
dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.
Shaw, Amy, "Games on Grids and Graphs" (2017). Electronic Theses and Dissertations. 1615.