Browsing Combinatorics and Optimization by Subject "graph theory"
Now showing items 1-20 of 33
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2-crossing critical graphs with a V8 minor
(University of Waterloo, 2012-01-17)The crossing number of a graph is the minimum number of pairwise crossings of edges among all planar drawings of the graph. A graph G is k-crossing critical if it has crossing number k and any proper subgraph of G has a ... -
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 ... -
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 ... -
Analytic Methods and Combinatorial Plants
(University of Waterloo, 2024-04-08)Combinatorial structures have broad applications in computer science, from error-correcting codes to matrix multiplication. Many analytic tools have been developed for studying these structures. In this thesis, we examine ... -
Analyzing Tree Attachments in 2-Crossing-Critical Graphs with a V8 Minor
(University of Waterloo, 2023-04-25)The crossing number of a graph is the minimum number of pairwise edge crossings in a drawing of the graph in the plane. A graph G is k-crossing-critical if its crossing number is at least k and if every proper subgraph H ... -
Aspects of Quantum Field Theory in Enumerative Graph Theory
(University of Waterloo, 2022-10-24)While a quantum field theorist has many uses for mathematics of all kinds, the relationship between quantum field theory and mathematics is far too fluid in the world of modern research to be described as the simple provision ... -
Clique minors in dense matroids
(University of Waterloo, 2022-09-23)The objective of this thesis is to bound the number of points a $U_{2,\ell+2}$- and $M(K_{k+1})$-minor-free matroid has. We first prove that a sufficiently large matroid will contain a structure called a tower. We then use ... -
Combinatorial Generalizations of Sieve Methods and Characterizing Hamiltonicity via Induced Subgraphs
(University of Waterloo, 2022-08-17)A sieve method is in effect an application of the inclusion-exclusion counting principle, and the estimation methods to avoid computing the explicit formula. Sieve methods have been used in number theory for over a hundred ... -
Combinatorially Thin Trees and Spectrally Thin Trees in Structured Graphs
(University of Waterloo, 2023-12-19)Given a graph $G=(V,E)$, finding simpler estimates of $G$ with possibly fewer edges or vertices while capturing some of its specific properties has been used in order to design efficient algorithms. The concept of estimating ... -
Constructing Cospectral and Comatching Graphs
(University of Waterloo, 2019-07-18)The matching polynomial is a graph polynomial that does not only have interesting mathematical properties, but also possesses meaningful applications in physics and chemistry. For a simple graph, the matching polynomial ... -
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, ... -
Density and Structure of Homomorphism-Critical Graphs
(University of Waterloo, 2018-08-22)Let $H$ be a graph. A graph $G$ is $H$-critical if every proper subgraph of $G$ admits a homomorphism to $H$, but $G$ itself does not. In 1981, Jaeger made the following conjecture concerning odd-cycle critical graphs: ... -
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 ... -
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 ... -
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 ... -
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 ... -
A Generalization to Signed Graphs of a Theorem of Sergey Norin and Robin Thomas
(University of Waterloo, 2019-12-19)In this thesis we characterize the minimal non-planar extensions of a signed graph. We consider the following question: Given a subdivision of a planar signed graph (G, Σ), what are the minimal structures that can be added ... -
Graph-Theoretic Techniques for Optimizing NISQ Algorithms
(University of Waterloo, 2024-02-15)Entering the NISQ era, the search for useful yet simple quantum algorithms is perhaps of more importance now than it may ever be in the future. In place of quantum walks, the quantum Fourier transform, and asymptotic results ... -
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 ... -
Local Perspectives on Planar Colouring
(University of Waterloo, 2022-08-09)In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erdos, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every ...