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Now showing items 1-10 of 18

#### Variations on a Theme: Graph Homomorphisms

(University of Waterloo, 2013-08-30)

This thesis investigates three areas of the theory of graph homomorphisms: cores of graphs, the homomorphism order, and quantum homomorphisms.
A core of a graph X is a vertex minimal subgraph to which X admits a ...

#### Hamilton Paths in Generalized Petersen Graphs

(University of Waterloo, 2002)

This thesis puts forward the conjecture that for <i>n</i> > 3<i>k</i> with <i>k</i> > 2, the generalized Petersen graph, <i>GP</i>(<i>n,k</i>) is Hamilton-laceable if <i>n</i> is even and <i>k</i> is odd, and it is ...

#### Core Structures in Random Graphs and Hypergraphs

(University of Waterloo, 2013-08-30)

The k-core of a graph is its maximal subgraph with minimum degree at least k. The study of k-cores in random graphs was initiated by Bollobás in 1984 in connection to k-connected subgraphs of random graphs. Subsequently, ...

#### Diameter and Rumour Spreading in Real-World Network Models

(University of Waterloo, 2015-04-20)

The so-called 'small-world phenomenon', observed in many real-world networks, is that there is a short path between any two nodes of a network, whose length is much smaller that the network's size, typically growing as a ...

#### On 2-crossing-critical graphs with a V8-minor

(University of Waterloo, 2014-05-22)

The crossing number of a graph is the minimum number of pairwise edge crossings in a drawing of a graph. A graph $G$ is $k$-crossing-critical if it has crossing number at least $k$, and any subgraph of $G$ has crossing ...

#### Morphing planar triangulations

(University of Waterloo, 2014-06-09)

A morph between two drawings of the same graph can be thought of as a continuous deformation between the two given drawings. A morph is linear if every vertex moves along a straight line segment from its initial position ...

#### The Erdős Pentagon Problem

(University of Waterloo, 2018-12-20)

The Erdős pentagon problem asks about the maximum number of copies of C_5 that one can find in a triangle-free graph. This problem was posed in 1984, but was not resolved until 2012. In this thesis, we aim to capture the ...

#### Algebraic Methods for Reducibility in Nowhere-Zero Flows

(University of Waterloo, 2007-09-25)

We study reducibility for nowhere-zero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counter-example to some conjecture. We derive algebraic proofs of ...

#### Properties of random graphs

(University of Waterloo, 2008-09-23)

The thesis describes new results for several problems in random graph theory.
The first problem relates to the uniform random graph model in
the supercritical phase; i.e. a graph, uniformly distributed, on $n$ vertices
and ...

#### Multigraphs with High Chromatic Index

(University of Waterloo, 2009-07-22)

In this thesis we take a specialized approach to edge-colouring by focusing exclusively on multigraphs with high chromatic index. The bulk of our results can be classified into three categories. First, we prove results ...