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Now showing items 1-10 of 12

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### Establishing a Connection Between Graph Structure, Logic, and Language Theory

(University of Waterloo, 2015-09-08)

The field of graph structure theory was given life by the Graph Minors Project of Robertson and Seymour, which developed many tools for understanding the way graphs relate to each other and culminated in the proof of the ...

#### Acyclic Colouring of Graphs on Surfaces

(University of Waterloo, 2018-09-04)

An acyclic k-colouring of a graph G is a proper k-colouring of G with no bichromatic cycles. In 1979, Borodin proved that planar graphs are acyclically 5-colourable, an analog of the Four Colour Theorem. Kawarabayashi and ...

#### FACES OF MATCHING POLYHEDRA

(University of Waterloo, 2016-09-30)

Let G = (V, E, ~) be a finite loopless graph, let
b=(bi:ieV) be a vector of positive integers. A
feasible matching is a vector X = (x.: j e: E)
J
of nonnegative
integers such that for each node i of G, the sum of ...

#### On Geometric Drawings of Graphs

(University of Waterloo, 2018-04-18)

This thesis is about geometric drawings of graphs and their topological generalizations.
First, we study pseudolinear drawings of graphs in the plane. A pseudolinear drawing is one in which every edge can be extended ...