Combinatorics and Optimization
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).
Waterloo faculty, students, and staff can contact us or visit the UWSpace guide to learn more about depositing their research.
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Discrete Quantum Walks on Graphs and Digraphs
(University of Waterloo, 20180926)This thesis studies various models of discrete quantum walks on graphs and digraphs via a spectral approach. A discrete quantum walk on a digraph $X$ is determined by a unitary matrix $U$, which acts on complex functions ... 
Results on Chromatic Polynomials Inspired by a Correlation Inequality of G.E. Farr
(University of Waterloo, 20180925)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 logconcave sequence. ... 
Ordinary and Generalized Circulation Algebras for Regular Matroids
(University of Waterloo, 20180919)Let E be a finite set, and let R(E) denote the algebra of polynomials in indeterminates (x_e)_{e in E}, modulo the squares of these indeterminates. Subalgebras of R(E) generated by homogeneous elements of degree 1 have ... 
NonConstructivity in Security Proofs
(University of Waterloo, 20180910)In the field of cryptography, one generally obtains assurances for the security of a cryptographic protocol by giving a reductionist security proof, which is comprised of a reduction from breaking a mathematical problem ... 
Acyclic Colouring of Graphs on Surfaces
(University of Waterloo, 20180904)An acyclic kcolouring of a graph G is a proper kcolouring of G with no bichromatic cycles. In 1979, Borodin proved that planar graphs are acyclically 5colourable, an analog of the Four Colour Theorem. Kawarabayashi and ... 
Density and Structure of HomomorphismCritical Graphs
(University of Waterloo, 20180822)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 oddcycle critical graphs: ... 
Induction Relations in the Symmetric Groups and JucysMurphy Elements
(University of Waterloo, 20180816)Transitive factorizations faithfully encode many interesting objects. The wellknown ones include ramified coverings of the sphere and hypermaps. Enumeration of specific classes of such objects have been known for quite ... 
Spectral properties of tensor products of channels
(Elsevier, 20180915)We investigate spectral properties of the tensor products of two completely positive and trace preserving linear maps (also known as quantum channels) acting on matrix algebras. This leads to an important question of when ... 
Action of degenerate Bethe operators on representations of the symmetric group
(University of Waterloo, 20180524)Degenerate Bethe operators are elements defined by explicit sums in the center of the group algebra of the symmetric group. They are useful on account of their relation to the GelfandZetlin algebra and the YoungJucysMurphy ... 
The Capacitated Matroid Median Problem
(University of Waterloo, 20180518)In this thesis, we study the capacitated generalization of the Matroid Median Problem which is a generalization of the classical clustering problem called the kMedian problem. In the capacitated matroid median problem, ... 
Extensions of Galvin's Theorem
(University of Waterloo, 20180430)We discuss problems in list coloring with an emphasis on techniques that utilize oriented graphs. Our central theme is Galvin's resolution of the Dinitz problem (Galvin. J. Comb. Theory, Ser. B 63(1), 1995, 153158). We ... 
Thin Trees in Some Families of Graphs
(University of Waterloo, 20180425)Let 𝐺=(𝑉,𝐸) be a graph and let 𝑇 be a spanning tree of 𝐺. The thinness parameter of 𝑇 denoted by 𝜌(𝑇) is the maximum over all cuts of the proportion of the edges of 𝑇 in the cut. Thin trees play an important role ... 
Ideal Clutters
(University of Waterloo, 20180424)Let E be a finite set of elements, and let C be a family of subsets of E called members. We say that C is a clutter over ground set E if no member is contained in another. The clutter C is ideal if every extreme point of ... 
On Geometric Drawings of Graphs
(University of Waterloo, 20180418)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 ... 
A Primal Dual Algorithm On 2Steiner Graphs
(University of Waterloo, 20180123)The Steiner Tree Problem is a fundamental network design problem, where the goal is to connect a subset of terminals of a given network at minimum cost. A major open question regarding this problem, is proving that the ... 
5Choosability of Planarplustwoedge Graphs
(University of Waterloo, 20180102)We prove that graphs that can be made planar by deleting two edges are 5choosable. To arrive at this, first we prove an extension of a theorem of Thomassen. Second, we prove an extension of a theorem Postle and Thomas. ... 
Extending Pappus' Theorem
(University of Waterloo, 20171222)Let $M_1$ and $M_2$ be matroids such that $M_2$ arises from $M_1$ by relaxing a circuithyperplane. We will prove that if $M_1$ and $M_2$ are both representable over some finite field $GF(q)$, then $M_1$ and $M_2$ have ... 
Linearlydense classes of matroids with bounded branchwidth
(University of Waterloo, 20170927)Let $M$ be a nonempty minorclosed class of matroids with bounded branchwidth that does not contain arbitrarily large simple rank$2$ matroids. For each nonnegative integer $n$ we denote by $ex(n)$ the size of the ... 
Combinatorial Arithmetic on Elliptic Curves
(University of Waterloo, 20170927)We propose a scalar multiplication technique on an elliptic curve, which operates on triples of collinear points. The computation of this operation requires a new approach to operation chains, with similarities to Montgomery ... 
On The Circuit Diameters of Some Combinatorial Polytopes
(University of Waterloo, 20170920)The combinatorial diameter of a polytope P is the maximum value of a shortest path between two vertices of P, where the path uses the edges of P only. In contrast to the combinatorial diameter, the circuit diameter of P ...