Electronic Theses and Dissertations

Identifier

238

Date

2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematical Sciences

Concentration

Mathematics

Committee Chair

Bela Bollobas

Committee Member

Paul Balister

Committee Member

Anna Kaminska

Committee Member

Jeno Lehel

Abstract

In this thesis, I examine two different problems in graph theory using probabilistic techniques. The first is a question on graph colourings. A proper total k-colouring of a graph G = (V, E) is a map φ : V υ E → {1, 2,…, k} such that φ|V is a proper vertex colouring, φ|E is a proper edge colouring, and if v V and vw E then φ(v) ≠ φ(vw). Such a colouring is called adjacent vertex distinguishing if for every pair of adjacent vertices, u and v, the set {φ(u)} υ {φ(uw) : uw E}, the `colour set of u', is distinct from the colour set of v. It is shown that there is an absolute constant C such that the minimal number of colours needed for such a colouring is at most Δ(G) + C. The second problem is related to a modification of bootstrap percolation on a finite square grid. In an n × n grid, the 1 × 1 squares, called sites, can be in one of two states: `uninfected' or `infected'. Sites are initially infected independently at random and the state of each vertex is updated simultaneously by the following rule: every uninfected site that shares an edge with at least two infected sites becomes itself infected while each infected site with no infected neighbours becomes uninfected. This process is repeated and the central question is, when is it either likely or unlikely that all sites eventually become infected? Here, both upper and lower bounds are given for the probability that all sites eventually become infected and these bounds are used to determine a critical probability for the event that all sites eventually become infected.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.

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