Browsing Combinatorics and Optimization by Subject "Graph Theory"
Now showing items 1-13 of 13
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Applied Hilbert's Nullstellensatz for Combinatorial Problems
(University of Waterloo, 2016-09-23)Various feasibility problems in Combinatorial Optimization can be stated using systems of polynomial equations. Determining the existence of a \textit{stable set} of a given size, finding the \textit{chromatic number} of ... -
Circle Graph Obstructions
(University of Waterloo, 2017-08-31)In this thesis we present a self-contained proof of Bouchet’s characterization of the class of circle graphs. The proof uses signed graphs and is analogous to Gerards’ graphic proof of Tutte’s excluded-minor characterization ... -
Cyclically 5-Connected Graphs
(University of Waterloo, 2016-08-29)Tutte's Four-Flow Conjecture states that every bridgeless, Petersen-free graph admits a nowhere-zero 4-flow. This hard conjecture has been open for over half a century with no significant progress in the first forty years. ... -
Equiangular Lines and Antipodal Covers
(University of Waterloo, 2010-09-22)It is not hard to see that the number of equiangular lines in a complex space of dimension $d$ is at most $d^{2}$. A set of $d^{2}$ equiangular lines in a $d$-dimensional complex space is of significant importance in Quantum ... -
Highly Non-Convex Crossing Sequences
(University of Waterloo, 2012-05-18)For a given graph, G, the crossing number crₐ(G) denotes the minimum number of edge crossings when a graph is drawn on an orientable surface of genus a. The sequence cr₀(G), cr₁(G), ... is said to be the crossing sequence ... -
Matchings and Covers in Hypergraphs
(University of Waterloo, 2016-05-17)In this thesis, we study three variations of matching and covering problems in hypergraphs. The first is motivated by an old conjecture of Ryser which says that if $\mcH$ is an $r$-uniform, $r$-partite hypergraph which ... -
Minimum Crossing Problems on Graphs
(University of Waterloo, 2007-01-19)This thesis will address several problems in discrete optimization. These problems are considered hard to solve. However, good approximation algorithms for these problems may be helpful in approximating problems in ... -
Minimum Number of Triangles of K5 Descendants
(University of Waterloo, 2022-01-26)In the study of Quantum Field Theory and Feynman Periods, the operation of double triangle expansion plays an important role. This is largely due to double triangle expansions not affecting the maximum weight of the ... -
Planar graphs without 3-cycles and with 4-cycles far apart are 3-choosable
(University of Waterloo, 2016-09-16)A graph G is said to be L-colourable if for a given list assignment L = {L(v)|v ∈ V (G)} there is a proper colouring c of G such that c(v) ∈ L(v) for all v in V (G). If G is L-colourable for all L with |L(v)| ≥ k for all ... -
Properties of graphs with large girth
(University of Waterloo, 2008-01-24)This thesis is devoted to the analysis of a class of iterative probabilistic algorithms in regular graphs, called locally greedy algorithms, which will provide bounds for graph functions in regular graphs with ... -
Properties of Stable Matchings
(University of Waterloo, 2010-12-17)Stable matchings were introduced in 1962 by David Gale and Lloyd Shapley to study the college admissions problem. The seminal work of Gale and Shapley has motivated hundreds of research papers and found applications in ... -
Results on Chromatic Polynomials Inspired by a Correlation Inequality of G.E. Farr
(University of Waterloo, 2018-09-25)In1993 Graham Farr gave a proof of a correlation inequality involving colourings of graphs. His work eventually led to a conjecture that number of colourings of a graph with certain properties gave a log-concave sequence. ... -
Uniform Mixing of Quantum Walks and Association Schemes
(University of Waterloo, 2013-09-26)In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time ...