Combinatorics and Optimization

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

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Browsing Combinatorics and Optimization by Author "Geelen, Jim"

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    Circle Graph Obstructions
    (University of Waterloo, 2017-08-31) Lee, Edward; Geelen, Jim
    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 of the class of graphic matroids.
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    Disasters in Abstracting Combinatorial Properties of Linear Dependence
    (University of Waterloo, 2020-05-15) Campbell, Rutger Theodoor Ronald Jansen van Doorn; Geelen, Jim
    A notion of geometric structure can be given to a set of points without using a coordinate system by instead describing geometric relations between finite combinations of elements. The fundamental problem is to then characterize when the points of such a “geometry” have a consistent coordinatization. Matroids are a first step in such a characterization as they require that geometric relations satisfy inherent abstract properties. Concretely, let E be a finite set and I be a collection of subsets of E. The problem is to characterize pairs (E,I) for which there exists a “representation” of E as vectors in a vector space over a field F where I corresponds to the linear independent subsets of E. Necessary conditions for such a representation to exist include: the empty set is independent, subsets of independent sets are also independent, and for each subset X, the maximal independent subsets of X have the same size. When these properties hold, we say that (E,I) describes a matroid. As a result of these properties, matroids provide many useful concepts and are an appropriate context in which to consider characterizations. Mayhew, Newman, and Whittle showed that there exist pathological obstructions to natural axiomatic and forbidden-substructure characterizations of real-representable matroids. Furthermore, an extension of a result of Seymour illustrates that there is high computational complexity in verifying that a representation exists. This thesis shows that such pathologies still persist even if it is known that there exists a coordinatization with complex numbers and a sense of orientation, both of which are necessary to have a coordinatization over the reals.
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    Extending Pappus' Theorem
    (University of Waterloo, 2017-12-22) Hoersch, Florian; Geelen, Jim
    Let $M_1$ and $M_2$ be matroids such that $M_2$ arises from $M_1$ by relaxing a circuit-hyperplane. 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 highly structured representations. Roughly speaking, $M_1$ and $M_2$ have representations that can be partitioned into a bounded number of blocks each of which is \enquote{triangular}, a property we call weakly block-triangular. Geelen, Gerards and Whittle have announced that, under the hypotheses above, the matroids $M_1$ and $M_2$ both have pathwidth bounded by some constant depending only on $q$. That result plays a significant role in their announced proof of Rota's Conjecture. Bounding the pathwidth of $M_1$ and $M_2$ is currently the single most complicated part in the proof of Rota's Conjecture. Our result is intended as a step toward simplifying this part. A matroid $N$ is said to be a fragile minor of another matroid $M$ if $M/C\backslash D = N$ for some $C,D \subseteq E(M)$, but $M/C'\backslash D' \neq N$ whenever $C \neq C'$ or $D \neq D'$. As a second result, we will prove that, given a $GF(q)$-representable matroid $N$, every $GF(q)$-representable matroid $M$ having $N$ as a fragile minor has a representation which is weakly block-triangular.
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    Linearly-dense classes of matroids with bounded branch-width
    (University of Waterloo, 2017-09-27) Hill, Owen; Geelen, Jim
    Let $M$ be a non-empty minor-closed class of matroids with bounded branch-width that does not contain arbitrarily large simple rank-$2$ matroids. For each non-negative integer $n$ we denote by $ex(n)$ the size of the largest simple matroid in $M$ that has rank at most $n$. We prove that there exist a rational number $\Delta$ and a periodic sequence $(a_0,a_1,\ldots)$ of rational numbers such that $ex(n) = \Delta n+a_n$ for each sufficiently large integer $n$.
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    Local properties of graphs with large chromatic number
    (University of Waterloo, 2022-08-31) Davies, James; Geelen, Jim
    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 $\mathbb{R}^3$ that have arbitrarily large girth and chromatic number. We also prove that the maximum chromatic number of a circle graph with clique number at most $\omega$ is equal to $\Theta(\omega \log \omega)$. Lastly, extending the $\chi$-boundedness of circle graphs, we prove a conjecture of Geelen that every proper vertex-minor-closed class of graphs is $\chi$-bounded.
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    Local Structure for Vertex-Minors
    (University of Waterloo, 2021-10-12) McCarty, Rose; Geelen, Jim
    This thesis is about a conjecture of Geelen on the structure of graphs with a forbidden vertex-minor; the conjecture is like the Graph Minors Structure Theorem of Robertson and Seymour but for vertex-minors instead of minors. We take a step towards proving the conjecture by determining the "local structure''. Our first main theorem is a grid theorem for vertex-minors, and our second main theorem is more like the Flat Wall Theorem of Robertson and Seymour. We believe that the results presented in this thesis provide a path towards proving the full conjecture. To make this area more accessible, we have organized the first chapter as a survey on "structure for vertex-minors''.
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    On the Excluded Minors for Dyadic Matroids
    (University of Waterloo, 2019-01-17) Wong, Chung-Yin; Geelen, Jim
    The study of the class of dyadic matroids, the matroids representable over both $GF(3)$ and $GF(5)$, is a natural step to finding the excluded minors for $GF(5)$-representability. In this thesis we characterize the ternary matroids $M$ that are excluded minors for dyadic matroids and contains a 3-separation. We will show that one side of the separation has size at most four, and that $M$ is obtained by adding at most four elements to another excluded minor $M'$. This reduces the problem of finding the excluded minors for dyadic matroids to the problem of finding the vertically 4-connected excluded minors for dyadic matroids.
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    Unavoidable Minors of Large 5-Connected Graphs
    (University of Waterloo, 2016-08-24) Shantanam, Abhinav; Geelen, Jim
    This thesis shows that, for every positive integer $n \geq 5$, there exists a positive integer $N$ such that every $5-$connected graph with at least $N$ vertices has a minor isomorphic to one of thirty explicitly defined $5-$connected graphs $H_1(n), ..., H_{30}(n)$, each with at least $n$ vertices.