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This is the collection for the University of Waterloo's Department of Combinatorics and Optimization .

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

### Recent deposits

• #### Implicit Loss of Surjectivity and Facial Reduction: Theory and Applications ﻿

(University of Waterloo, 2023-03-09)
Facial reduction, pioneered by Borwein and Wolkowicz, is a preprocessing method that is commonly used to obtain strict feasibility in the reformulated, reduced constraint system. The importance of strict feasibility is ...
• #### The Edmonds-Giles Conjecture and its Relaxations ﻿

(University of Waterloo, 2022-12-23)
Given a directed graph, a directed cut is a cut with all arcs oriented in the same direction, and a directed join is a set of arcs which intersects every directed cut at least once. Edmonds and Giles conjectured for all ...
• #### k-Connectedness and k-Factors in the Semi-Random Graph Process ﻿

(University of Waterloo, 2022-12-20)
The semi-random graph process is a single-player graph game where the player is initially presented an edgeless graph with n vertices. In each round, the player is offered a vertex u uniformly at random and subsequently ...
• #### Mixed Integer Programming Approaches for Group Decision Making ﻿

(University of Waterloo, 2022-10-26)
Group decision making problems are everywhere in our day-to-day lives and have great influence on the daily operation of companies and institutions. With the recent advances in computational technology, it's not surprising ...
• #### 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 ...
• #### Digraphs with All Induced Directed Cycles of the Same Length are not → χ -Bounded ﻿

(2022-10-07)
For t > 2, let us call a digraph D t-chordal if all induced directed cycles in D have length equal to t. In an earlier paper, we asked for which t it is true that t-chordal graphs with bounded clique number have bounded ...
• #### Enumerative perspectives on chord diagrams ﻿

(University of Waterloo, 2022-10-07)
The topic of this thesis is enumerating certain classes of chord diagrams, perfect matchings of the interval $\{1, 2, \ldots, 2n\}$. We consider hereditary classes of chord diagrams that are restricted to satisfy one of ...
• #### Edge-disjoint Linkages in Infinite Graphs ﻿

(University of Waterloo, 2022-09-26)
The main subject of this thesis is the infinite graph version of the weak linkage conjecture by Thomassen [24]. We first prove results about the structure of the lifting graph; Theorems 2.2.8, 2.2.24, and 2.3.1. As an ...
• #### 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 ...
• #### Transversal Problems In Sparse Graphs ﻿

(University of Waterloo, 2022-09-20)
Graph transversals are a classical branch of graph algorithms. In such a problem, one seeks a minimum-weight subset of nodes in a node-weighted graph $G$ which intersects all copies of subgraphs~$F$ from a fixed family ...
• #### A Counterexample to a Conjecture About Trianle-Free Induced Subgraphs of Graphs with Large Chromatic Number ﻿

(Elsevier ScienceDirect, 2023-01)
We prove that for every n, there is a graph G with χ(G) ≥ n and ω(G) ≤ 3 such that every induced subgraph H of G with ω(H) ≤ 2 satisfies χ(H) ≤ 4.This disproves a well-known conjecture. Our construction is a digraph with ...
• #### Induced Subgraphs and Tree Decompositions III. Three-Path-Configurations and Logarithmic Treewidth. ﻿

A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family H of graphs, we say a graph G is H-free if no induced subgraph of G is ...
• #### Two combinatorial problems from craniosynostosis ﻿

(University of Waterloo, 2022-09-08)
The art of cranial vault remodelling surgery is amazing and fascinating, but remains much of that, an art, to this day. In this thesis, we provide two mathematical approaches to tackle cranial vault remodelling surgery, ...
• #### Polyhedral Diameters and Applications to Optimization ﻿

(University of Waterloo, 2022-09-01)
The Simplex method is the most popular algorithm for solving linear programs (LPs). Geometrically, it moves from an initial vertex solution to an improving neighboring one, selected according to a pivot rule. Despite decades ...
• #### On the Power and Limitations of Shallow Quantum Circuits ﻿

(University of Waterloo, 2022-09-01)
Constant-depth quantum circuits, or shallow quantum circuits, have been shown to exhibit behavior that is uniquely quantum. This thesis explores the power and limitations of constant-depth quantum circuits, in particular ...
• #### Complexity Dichotomy for List-5-Coloring with a Forbidden Induced Subgraph ﻿

(Society for Industrial and Applied Mathematics, 2022-08-30)
For a positive integer r and graphs G and H, we denote by G+H the disjoint union of G and H and by rH the union of r mutually disjoint copies of H. Also, we say G is H-free if H is not isomorphic to an induced subgraph of ...
• #### Local properties of graphs with large chromatic number ﻿

(University of Waterloo, 2022-08-31)
This thesis deals with problems concerning the local properties of graphs with large chromatic number in hereditary classes of graphs. We construct intersection graphs of axis-aligned boxes and of lines in $\mathbb{R}^3$ ...
• #### Algorithm Design for Ordinal Settings ﻿

(University of Waterloo, 2022-08-29)
Social choice theory is concerned with aggregating the preferences of agents into a single outcome. While it is natural to assume that agents have cardinal utilities, in many contexts, we can only assume access to the ...
• #### Lattice Paths ﻿

(University of Waterloo, 2022-08-24)
This thesis is a survey of some of the well known results in lattice path theory. Chapter 1 looks into the history of lattice paths. That is, when it began and how it was popularized. Chapter 3 focuses on general lattices ...
• #### Cycles and coloring in graphs and digraphs ﻿

(University of Waterloo, 2022-08-22)
We show results in areas related to extremal problems in directed graphs. The first concerns a rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture, made by Aharoni. The Caccetta-H\"{a}ggkvist conjecture states ...

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