Combinatorics and Optimization

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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).

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Preserving and Generalizing χ-boundedness
(University of Waterloo, 2025-10-27) Chaniotis, Aristotelis
The notion of χ-boundedness, introduced by Gyárfás in the mid-1980s, captures when, for every induced subgraph of a graph, large chromatic number can occur only due to the presence of a sufficiently large complete subgraph. The study of χ-boundedness is a central topic in graph theory. Understanding which hereditary classes of graphs are χ-bounded is of particular importance for advancing our understanding of how restrictions on the induced subgraphs of a graph affect both its global structure and key parameters such as the clique number and the independence number. Which classes of graphs are χ-bounded? A method that has been used to prove that a class C of graphs is χ-bounded proceeds as follows: we prove that C can be obtained by applying operations that preserve χ-boundedness to already χ-bounded classes. This approach gives rise to the following question: which operations preserve χ-boundedness? Given k graphs G₁,…,Gₖ, their intersection is the graph (∩{i∈[k]}V(Gᵢ), ∩{i∈[k]}E(Gᵢ)). Given k graph classes G₁,…,Gₖ, we call the class {G : ∀i∈[k], ∃Gᵢ∈Gᵢ such that G = G₁ ∩ ⋯ ∩ Gₖ} the graph-intersection of G₁,…,Gₖ. In the mid-1980s, in his seminal paper “Problems from the world surrounding perfect graphs”, Gyárfás observed that, due to early results of Asplund and Grünbaum, and Burling, graph-intersection does not preserve χ-boundedness in general, and he raised some questions regarding the interplay between graph-intersection and χ-boundedness. This topic has not received much attention since then. In this thesis, we formalize and explore the connection between the operation of graph-intersection and χ-boundedness. Let r ≥ 2 be an integer. We denote by Kᵣ the complete graph on r vertices. The Kᵣ-free chromatic number of a graph G, denoted by χᵣ(G), is the minimum size of a partition of V(G) into sets each of which induces a Kᵣ-free graph. Generalizing χ-boundedness, we say that a class C of graphs is χᵣ-bounded if there exists a function f:ℕ→ℕ such that for every G∈C and every induced subgraph G′ of G, we have χᵣ(G′) ≤ f(ω(G′)), where ω(G′) denotes the clique number of G′. We study the induced subgraphs of graphs with large Kᵣ-free chromatic number. Finally, we introduce the fractional Kᵣ-free chromatic number, and for every r ≥ 2 we construct K_{r+1}-free intersection graphs of straight-line segments in the plane with arbitrarily large fractional Kᵣ-free chromatic number.
Forbidding odd K3,3 as a graft minor
(University of Waterloo, 2025-10-06) Reinert, Nathan
A graph is odd−K5 free if K5 cannot be obtained by deleting edges and then contracting all edges in a cut. odd − K5 free graphs play an important role in the study of multi-commodity flows. A graph is odd − K3,3 free if K3,3 cannot be obtained by contracting edges and then deleting all edges in an eulerian subgraph. A long-standing conjecture of Paul Seymour predicts that postman sets pack in odd − K3,3 free graphs. We study odd − K3,3 free graphs that are almost planar in this thesis and discuss the relation to Seymour’s conjecture.
Approximation Algorithms for Relative Survivable Network Design Problems
(University of Waterloo, 2025-09-19) Nan, JJ
The Survivable Network Design (SND) problem is a classical and well-studied graph connectivity problem. Given a set of source-sink pairs and demands between them, SND asks one to compute a subgraph such that the number of paths between each pair meets their demand. SND is primarily interesting in modeling fault-tolerance; we can see the problem as requiring certain nodes to be connected even if some edges ”fail”. It is well known that a 2-approximation algorithm for SND exists, using the method of iterative rounding. In 2022, Dinitz et al. introduced a problem that we refer to as Path-Relative Survivable Network Design (PRSND), a natural extension of SND that addresses cases where the underlying graph does not have the required connectivity; in this problem, we require that the connectivity of our subgraph is ”as good as” it is in the original graph. Perhaps surprisingly, this variation makes PRSND much harder to approximate than standard SND, and outside of certain special cases no constant-factor approximations have been found. In this thesis we introduce the Cut-Relative Survivable Network Design (CRSND) prob- lem, another variant of SND that similarly aims to capture relative fault-tolerance. We show that this problem admits a 2-approximation algorithm, matching the best known ap- proximation factor for SND, via a decomposition technique. We explore some properties of said approximation, as well as hardness and modeling properties of Cut-Relative Network Design problems.
Cuts in Optimization and Approximation: Generalized MIR Cutting Plane and Integrality Gap Bounds for the Planar Multicut Problem
(University of Waterloo, 2025-09-19) Kalantarzadeh, Sina
This thesis has two main themes, both centered around the role of cuts in integer programming and approximation algorithms. In the first part, we investigate the complexity of split closures in mixed-integer sets in~$\mathbb{R}^3$ with two integral variables and one continuous variable. While it is known that in~$\mathbb{R}^2$ the split closure admits a polynomial-size description, extending this result to the mixed-integer setting in~$\mathbb{R}^3$ poses new challenges. For a rational polyhedron~$P \subseteq \mathbb{R}^3$ and integer index set~$I = \{1,2\}$, we denote by~$P_I := conv(P \cap (\mathbb{Z}^2 \times \mathbb{R}))$ the mixed-integer hull of $P$ with respect to $I$, and by~$P^{split}_I$ the intersection of all split cuts with respect to $I$. We make progress toward understanding the structure of~$P^{split}_I$ by proving the existence of a polyhedron~$Q$ such that~$P_I \subseteq Q \subseteq P^{split}_I$, where~$Q$ can be described by a polynomial number of inequalities relative to the input size of~$P$. In addition, we introduce a generalized mixed-integer rounding (MIR) procedure that begins with systems of two equations, rather than a single equation as in the classical setting. This leads to a new family of valid inequalities for mixed-integer sets, whose derivation relies on structural results for split closures in~$\mathbb{R}^3$. The second part focuses on approximation algorithms for the minimum multicut problem in planar graphs and certain subclasses thereof. We analyze the integrality gap of the natural LP relaxation within the framework of small-diameter decompositions. By introducing novel tools, we improve the best known lower bound on this integrality gap from $2$ to $\tfrac{20}{9}$ for the family of cactus graphs, which in turn yields an improved lower bound of the integrality gap for planar graphs. Complementing this, we establish an upper bound of $3.44$ for cactus graphs, improving the previous bound of $4$. We further obtain tight bounds in special subclasses: the integrality gap is exactly $2$ for unicyclic graphs and for path cactus graphs. Extending the study to outerplanar graphs, we prove an upper bound of $8$ on the integrality gap by designing a randomized rounding algorithm that transforms the optimal fractional solution of the LP relaxation into a feasible multicut whose expected cost is at most $8$ times the cost of the optimal fractional solution. Together, these results sharpen our understanding of the minimum multicut problem and its LP relaxation in planar graphs.
Cover Small Cuts and Flexible Graph Connectivity Problems
(University of Waterloo, 2025-09-15) Simmons, Miles
Given a graph with capacitated edges and a set of links on the same vertex set, the Cover Small Cuts problem aims to choose a minimum-cost link set such that for each cut with total edge capacity below a given threshold, at least one selected link covers that cut. We present examples and analysis of the Cover Small Cuts problem, and cut covering problems on strongly pliable families, a subfamily of pliable families. Pliable set families require that for any two sets A, B in the family, at least two of A-B, B-A, A∪B, A∩B are also in the family. Strongly pliable set families require that for any two sets A, B in the family that cross, at least one of A-B, B-A is also in the family, and at least one of A∪B, A∩B is also in the family. Shortly before the submission of this thesis, we learned about prior work on what we call strongly pliable families; see Section 1.4.1 for further details. We decided to stay with the term "strongly pliable families" since it is consistent with the notion of "pliable families" introduced recently. We present a family of Cover Small Cuts problems such that key property used to prove the approximation ratio of Jain's iterative rounding approximation algorithm does not hold. We show that families of “small cuts” are always strongly pliable, but not all strongly pliable families can be realized as families of small cuts. We prove a 5-approximation algorithm for cut covering problems on strongly pliable families using the primal-dual method of Williamson, Goemans, Vazirani, and Mihail. We compare strongly pliable families to other subfamilies of pliable families, such as uncrossable families. We also study some other cut covering problems. In Flexible Graph Connectivity (FGC) problems, the edges of the input graph are partitioned into safe and unsafe edges. In real-world applications, safe edges represent reliable connections and unsafe edges represent connections that could break. We aim to select a minimum-cost set of edges that achieve given connectivity requirements despite the limitations of the unsafe edges. We present a constant-factor approximation algorithm for the (1, q)-FGC problem. We explore relaxations of the (p, q)-FGC problem that allow multiple copies of an edge to be selected, and construct approximation algorithms with better approximation ratios than those currently known for the (unrelaxed) (p, q)-FGC problem. We end with a summary of Williamson et al.’s primal-dual approximation algorithm, a versatile approximation algorithm for cut covering problems including Cover Small Cuts; and our results from computational experiments on this algorithm.
Isogeny-Based Zero-Knowledge Proofs and Their Applications
(University of Waterloo, 2025-09-15) Mokrani, Youcef
Isogeny-based cryptography is one of the main avenues of research in post-quantum cryptography. The fundamental idea of this breach is that there is currently no known efficient algorithm to compute an isogeny between two supersingular elliptic curves, even when one has access to a quantum computer. However, this pure primitive keeps too much information about the secret isogeny hidden to be directly applied to most applications. As such, almost every protocol based on isogeny reveals some extra information about the secret isogeny. This is famously the case for Supersingular Isogeny Diffie--Hellman (SIDH), which transmits the mapping of the isogeny on a torsion subgroup, the degree of the isogeny, and the endomorphism ring of the domain curve. The recent polynomial-time attacks on SIDH have shown that leaking the torsion subgroup mapping gives away too much information to an attacker. Because of this, the SIDH variants proposed to resist these attacks all mask the mapping in some way. However, less attention has been paid to the other types of information that SIDH and most of its new variants transmit. This is especially worrying when it comes to the endomorphism ring of the starting curve, as it was shown multiple times that it can lead to easier attacks. In fact, the first of the recent polynomial-time attacks on SIDH made direct use of the endomorphism ring. Also, before these attacks fully broke SIDH, Petit showed that knowledge of the endomorphism ring could lead to a polynomial-time attack on SIDH when the parameter sets were unbalanced. Castryck and Vercauteren recently showed that the same attacks on unbalanced parameters with known endomorphism rings can be extended to some of the new SIDH variants, such as M-SIDH. An interesting fact about SIDH variants is that most of them do not explicitly need to transmit the endomorphism ring. Most implementations do so because it is simpler. The goal of this thesis is therefore to further study the case for masking the endomorphism ring of the domain curve for SIDH variants. We start by showing that, for well-chosen parameter sets, working with a random starting curve can never lead to a loss of security. This thesis also explores the use of multiparty computations to generate curves of unknown endomorphism rings. Finally, we present a new set of zero-knowledge proofs for SIDH variants that do not require knowledge of any endomorphism rings and can be made to mask the degree of the secret isogeny.
Reflected and nonsymmetric crystal graphs
(University of Waterloo, 2025-09-05) Niergarth, Harper
This thesis is comprised of two projects. The first studies a certain composition of crystal operators on semistandard Young tableaux, which we term raised reflection operators, and are related to a sign-reversing involution used to prove the Littlewood--Richardson rule. In particular, we investigate the graph defined by these crystal operators. Our main result is that this graph is balanced bipartite, giving another proof of the Littlewood--Richardson rule. We do so by giving a set of local rules that this graph satisfies and showing that any graph satisfying these rules is balanced bipartite. The second studies crystal operators on multiline queues. Certain multiline queues, called non-wrapping multiline queues, are in bijection with semistandard Young tableaux but are better equipped to study nonsymmetric polynomials called Demazure atoms. Indeed, each multiline queue comes equipped with a weak composition, called the type, and summing over all multiline queues of a fixed type yields a Demazure atom. Crystal operators on multiline queues do not preserve type. Our main result characterizes how crystal operators interact with the type of a multiline queue. In particular, we show that these operators may only change the type by a simple transposition and that the type changes if and only if the multiline queue is in a specific configuration.
Acyclic List Colouring Locally Planar Graphs
(University of Waterloo, 2025-08-25) Vicenzo, Massimo
A (vertex) colouring of a graph is \emph{acyclic} if it contains no bicoloured cycle. In 1979, Borodin proved that planar graphs are acyclically 5-colourable. In 2010, Kawarabayashi and Mohar proved that locally planar graphs are acyclically 7-colourable. In 2002, Borodin, Fon-Der-Flaass, Kostochka, Raspaud, and Sopena proved that planar graphs are acyclically 7-list-colourable. We prove that locally planar graphs are acyclically 9-list-colourable - no bound for acyclic list colouring locally planar graphs for any fixed number of colours was previously known. We further show that triangle-free locally planar graphs are acyclically 8-list-colourable.
Multivariate Limit Theorems and Algebraic Generating Functions
(University of Waterloo, 2025-08-22) Ruza, Tiadora Valentina
The field of analytic combinatorics is dedicated to the creation of effective techniques to study the large-scale behaviour of combinatorial objects. Although classical results in analytic combinatorics are mainly concerned with univariate generating functions, over the last two decades a theory of Analytic Combinatorics in Several Variables (ACSV) has been developed to study the asymptotic behaviour of multivariate sequences. This thesis provides results for two areas of ACSV: limit theorems and asymptotics of algebraic generating functions. For both, the aim is to provide readers a blueprint to apply the powerful tools of ACSV in their own work, making them more accessible to combinatorialists, probabilists, and those in adjacent fields. First, we survey ACSV from a probabilistic perspective, illustrating how its most advanced methods provide efficient algorithms to derive limit theorems, and comparing the results to past work deriving limit theorems. Using the results of ACSV, we provide a SageMath package that can automatically compute (and rigorously verify) limit theorems for a large class of combinatorial generating functions. To illustrate the techniques involved, we also establish explicit local central limit theorems for a family of combinatorial classes whose generating functions are linear in the variables tracking each parameter. Applications covered by this result include the distribution of cycles in certain restricted permutations (proving a limit theorem conjectured in work of Chung et al.), integer compositions, and n-colour compositions with varying restrictions and values tracked. Key to establishing these explicit results in an arbitrary dimension is an interesting symbolic determinant, which we compute by conjecturing and then proving an appropriate LU-factorization. The second part of this thesis shifts focus to the calculation of asymptotics of multivariate algebraic generating functions through ACSV. So far, the methods of ACSV have largely focused on rational (or, more generally, meromorphic) generating functions, although many natural combinatorial objects have generating functions with algebraic singularities. In this part, we survey techniques for analyzing multivariate algebraic generating functions, going into detail specifically for the process of embedding an algebraic generating function into a sub-series of a rational function of more variables. Other methods mentioned include explicit singularity analysis of algebraic singularities, and manipulation of complex integrals over algebraic hypersurfaces. We give implementations of the embedding techniques in the SageMath computer algebra system, and provide examples from the combinatorics literature.
Extensions of the Tutte Polynomial and Results on the Interlace Polynomial
(University of Waterloo, 2025-08-14) Reynes, Josephine
In graph theory, graph polynomials are an important tool to encode information from a graph. The Tutte polynomial, first introduced in 1947, is one of the most important graph polynomials due to its universality. Here, we present three classic definitions of the Tutte polynomial via a deletion-contraction recursion, via rank and nullity, and via activities. We will touch on the significance of this polynomial to the field of mathematics to motivate an extension to signed graphs. Extending the polynomial to retain deletion contraction and inactivity information, we introduce an extended Tutte polynomial to allow for the construction of a Tutte like polynomial on signed graphs. Using the extended information, we examine the monomials of these polynomials as grid walks. Using grid walking and the extended Tutte polynomial, we investigate the relationship between the Tutte polynomial of a graph and that of its bipartite representation. This is done with a view toward the construction of a Tutte like polynomial for oriented hypergraphs. While many graph polynomials are directly related to the Tutte polynomial, there are also a wide variety of polynomials related in special cases only. One such polynomial is the Martin polynomial and, related to it, the interlace polynomial. Here, we discuss how these two polynomials are related and how results on the Martin polynomial can be extended to the interlace polynomial. The Martin invariant, a specific evaluation of the Martin polynomial, obeys the symmetries of the Feynman period. The Feynman period of a graph is useful in quantum field theory, but difficult to compute and thus there is interest in finding graph invariants that have the same symmetries. It was established that the interlace polynomial on interlace graphs was equal to the Martin polynomial on the associated 4-regular graph. While only graphs that do not contain a set of forbidden vertex minors are interlace graphs, the interlace polynomial is defined over all graphs. We discuss how this provides a way to try and extend the notion of Feynman symmetries via the interlace polynomial and some specific classes of graphs with formulas. Additionally, the interlace polynomial is only equal to the Martin polynomial for interlace graphs of 4-regular graphs, but the Martin polynomial is defined for 2k-regular graphs. Thus, we work toward creating an interlace-like polynomial for graphs derived from 2k-regular cases of the Martin polynomial.
Combinatorial Aspects of Feynman Integrals and Causal Set Theory
(University of Waterloo, 2025-05-23) Shaban, Kimia
This thesis consists of two distinct sections, the first highlighting causal set theory (CST) and the second focusing on Feynman period estimation. This work specifically covers how discrete structures from algebraic combinatorics can be applied to problems from physics, and how we can use computational tools and techniques to help solve these problems from a mathematical perspective. In the first part of the thesis, we introduce CST, an approach to quantum gravity that focuses on how events in spacetime are causally related to one another. In CST, discretized spacetime is given by a locally finite poset. A significant portion of this section focuses on covtree, which allows us to evolve such a discrete spacetime, in a way that does not depend on arbitrary labelling. However recognizing nodes of covtree involve solving a particular downset reconstruction problem. To attack this we define a graph that compares downsets that differ by exactly one element, with a particular focus on the order dimension two case. This section of the thesis sheds light on evolving a spacetime by constructing future elements within the covtree framework. Reconstructing spacetime in this way will allow for researchers to advance our understanding of covtree’s structure and improves causal set theory as an approach to quantum gravity. In the second part, we provide an introduction to Feynman periods, explaining their significance in quantum field theory (QFT) calculations. We will discuss how machine learning models, such as linear and quadratic regression, and graph neural networks, can be applied to Feynman graphs and their properties, to predict the Feynman period. Even simple techniques like linear regression, on graph parameters only, which do not consider the graph structure directly, can make highly accurate predictions of Feynman periods. The work presented in this section, done jointly with Dr. Paul-Hermann Balduf, is published in the Journal of High Energy Physics. This research has significant implications for QFT computations, which can become computationally infeasible at large scales, and facilitates further exploration in particle physics.
Analysis of the Three-operator Davis-Yin Splitting in the Inconsistent Case
(University of Waterloo, 2025-05-20) Naguib, Andrew
This thesis analyzes the Davis–Yin three-operator splitting method in the inconsistent case, where the underlying monotone inclusion problem may fail to have a solution. The Davis–Yin algorithm extends the Douglas–Rachford and forward–backward splitting methods and is effective in reformulating optimization and inclusion problems as fixed-point iterations. Our study investigates its behavior when no fixed point exists. We prove, under mild assumptions, that the Davis–Yin shadow sequence converges to a solution of the normal problem, which represents a minimal perturbation of the original formulation.
Edge-Coloring Planar Graphs and the Cycling Conjecture
(University of Waterloo, 2025-05-20) Bourla, Gabriela
An r-graph is defined to be a graph where each vertex has degree r and any odd subset has at least r edges leaving it. This thesis focuses on the conjecture that any planar r-graph can be edge-colored with r colors. Past work has shown that the conjecture holds for r ≤ 8, but it becomes more difficult with each increase of r. We consider what occurs when r is very large. The main ideas of the thesis work with a minimal counterexample graph that is one of the smallest graphs to contradict the conjecture for a given r. To make a minimal counterexample easier to work with, we generalize to grafts, working with T-joins and T-cuts. We go through various directions to approach the problem and show properties of a minimal counterexample as well as questions that stand in the way of proving it does not exist.
Quantum programming and synthesis: Internalizing Clifford operations and beyond
(University of Waterloo, 2025-04-29) Winnick, Samuel
Clifford operations are a subset of quantum operations used extensively in quantum error correction and classical simulation of quantum circuits. The first part of the thesis is motivated by the problem of programming with generalized Clifford operations, such as the quantum Fourier transform. We delve into the algebraic complications that arise for systems of even dimension $d$, and we are particularly interested in the case when $d$ is a power of $2$. We apply our results in the design of a quantum functional programming language, in which the user does not have to worry about these irrelevant complications. Later, we consider the problem of compiling circuits over universal gate sets. In particular, we study the problem of multi-qutrit exact synthesis over a variety of gate sets including Clifford gates. Lastly, we present a framework for defining a symplectic form on an object in a sufficiently structured category, and lay out the theory, generalizing the theory of symplectic forms on a finite dimensional vector space or locally compact abelian group. In the process, we develop new results and perspectives on operations defined on categories.
Asymptotics of the number of lattice points in the transportation polytope via optimization on Lorentzian polynomials.
(University of Waterloo, 2025-04-28) Lee, Thomas
We formally extend the theory of polynomial capacity to power series and totally uni- modular matrices. Using these results, we prove the log-asymptotic correctness of bounds by Brändén, Leake, and Pak developed through the use of Lorentzian polynomials ([BLP23]) under certain conditions, and provide a counterexample where these bounds are not log- asymptotically correct, even when symmetry exists.
Some Sylvester-Gallai-Type Theorems for Higher-Dimensional Flats
(University of Waterloo, 2025-04-28) Kroeker, Matthew Eliot
In response to an old question of Sylvester, Gallai proved that, for any finite set of points in the plane, not all contained in a single line, there is a line containing exactly two of them. This is known as the Sylvester-Gallai Theorem. In the language of matroid theory, which is used in this thesis, the Sylvester-Gallai Theorem says that every rank-3 real-representable matroid has a two-point line. Independently of Gallai and around the same time, Melchior proved the stronger result that, in a rank-3 real-representable matroid, the average number of points in a line is less than three. The purpose of this thesis is to prove theorems of this type for flats of higher dimension. The Sylvester-Gallai Theorem was generalized to higher dimensions by Hansen, who proved that, for any positive integer k, a simple real-representable matroid of rank at least 2k-1 has a rank-k independent flat. We will prove a theorem of this type for matroids representable over any given finite field by characterizing the "unavoidable flats" for matroids in this class. In particular, we show that any simple matroid of sufficiently high rank representable over a given finite field has a rank-k flat which is either independent or a projective or affine geometry. As a corollary, we will get the following Ramsey theorem: for any two-colouring of the points of a matroid of sufficiently high rank representable over a given finite field, there is a monochromatic rank-k flat. The rest of the thesis concerns generalizing Melchior’s theorem to higher dimensions. For planes, we prove that, in a real-representable matroid M of rank at least four, the average plane-size is at most an absolute constant, unless M is a direct sum of lines (in which case the average plane-size could be arbitrarily large). Generalizing this result to hyperplanes of arbitrary rank k, we prove that, if M is a simple rank-(k+1) real-representable matroid whose average hyperplane-size is greater than some constant depending only on k, then all but a constant number b_k of the points of M are covered by a small collection of flats of relatively low rank. We also show that this constant b_k is best-possible. Melchior’s Theorem also implies that the average size of a flat (of any rank) in a rank-3 real-representable matroid is at most an absolute constant. We prove that, for any positive integer r, the average size of a flat in a rank-r real-representable matroid is at most some constant depending only on r.
Denormalized Lorentzian Laurent series
(University of Waterloo, 2025-04-24) Mohammadi Yekta, Maryam
We introduce and study the class of denormalized Lorentzian Laurent series and prove lower bounds on their coefficients. To do this, we define a capacity function on Laurent series using their domain of convergences. As a result, we can use simpler arguments to re-prove an already known lower bound on the number of contingency tables, which are lattice points of transportation polytopes; and tidy up and slightly improve an already known lower bound on the number of integral flows of a network.
Max-Min Greedy Matching
(University of Waterloo, 2024-12-17) Rakhmatullin, Ramazan; Pashkovich, Kanstantsin
We study the problem of max-min greedy matching introduced in (Eden et al. 2022). The max-min greedy matching problem is a variant of the well known online bipartite matching problem introduced in (Karp et al. 1990). Given a bipartite graph $G=(U \cup V; E)$, the first player selects a priority ordering $\pi$ of $V$. After that the second player selects an arrival order $\sigma$ of $U$ depending on the ordering $\pi$ selected by the first player. Once both players did their selections, the vertices of $U$ arrive in the order specified by $\sigma$. Upon arrival, each vertex in $U$ is matched with its highest-priority neighbor that is still unmatched, if one exists. Here, the highest-priority is defined with respect to $\pi$. The first player aims to maximize the size of the resulting matching, while the second player seeks to minimize it. We investigate the max-min greedy matching problem. We aim to improve the lower bound of the guaranteed matching size achievable by the first player. We present a linear time algorithm that constructs an ordering $\pi$ such that for every $\sigma$ the ratio between the resulting matching size and the maximum matching size is at least $\frac{5}{9}$. Our result improves the previous best known ratio of $\frac{1}{2} + \frac{1}{86}$ from (Eden et al. 2022). The best known upper bound on the ratio is $\frac{2}{3}$ (Cohen-Addad et al. 2016). Thus, our result makes the gap between the lower and upper bounds for the ratio substantially smaller. Our method introduces a novel ``balanced path decomposition'' technique. This decomposition provides crucial structural properties such as existence of large matchings within these paths; and absence of certain edges between the paths. These structural properties allow us to prove the desired lower bound on the ratio. Our findings suggest that the achieved matching ratio can be further improved and that the balanced path decomposition may have broader applications.
On the relation among the maximum degree, the chromatic number and the clique number
(University of Waterloo, 2024-12-12) Xie, Xinyue; Haxell, Penny
We study the relation among the maximum degree $\Delta(G)$, the chromatic number $\chi(G)$ and the clique number $\omega(G)$ of a graph $G$ by proving results related to the Borodin-Kostochka Conjecture and Reed's Conjecture, which are two long-standing conjectures on this subject. The Borodin-Kostochka Conjecture states that a graph $G$ with $\chi(G)=\Delta(G)\ge9$ has a clique of size $\Delta(G)$. Reed's Conjecture states that any graph $G$ satisfies $\chi(G)\le \lceil\frac{1}{2}(\Delta(G)+\omega(G)+1)\rceil$. By combining and extending the approaches taken by Cranston and Rabern and by Haxell and MacDonald, we use the technique of Mozhan partitions to prove a generalized Borodin-Kostochka-type result as follows. Given a nonnegative integer $t$, for every graph $G$ with $\Delta(G)\ge 4t^2+11t+7$ and $\chi(G)=\Delta(G)-t$, the graph $G$ contains a clique of size $\Delta(G)-2t^2-7t-4$. We further prove that both conjectures hold for graphs with at least one critical vertex that is not in an odd hole, by following similar proofs as by Aravind, Karthick and Subramanian and by Chen, Lan, Lin and Zhou. This result then motivates us to study Reed's Conjecture in the context of graphs in which all vertices are in some odd hole. By investigating blow-ups and powers of cycles, we enumerate several classes of graphs for which Reed's Conjecture holds with equality. They generalize many of the known tight examples. We also introduce a class of graphs constructed from blow-ups and powers of paths, which gives rise to a new family of irregular tight examples.
Algebraic Approach to Quantum Isomorphisms
(University of Waterloo, 2024-09-24) Sobchuk, Mariia; Godsil, Chris
In very brief, this thesis is a study of quantum isomorphisms. We have started with two pairs of quantum isomorphic graphs and looked for generalizations of those. We have learned that those two pairs of graphs are related by Godsil-McKay switching and one of the graphs is an orthogonality graph of lines in a root system. These two observations lead to research in two directions. First, since quantum isomorphisms preserve coherent algebras, we studied a question of when Godsil-McKay switching preserved coherent algebras. In this way, non isomorphic graphs related by Godsil-McKay switching with isomorphic coherent algebras are candidates to being quantum isomorphic and non isomorphic. Second, while it was known that one of the graphs in a pair was an orthogonality graphs of the lines in a root system $E_8,$ we showed that a graph from another pair is also an orthogonality graph of the lines in a root system $F_4.$ We have studied orthogonality graphs of lines in root systems $B_{2^d},C_{2^d},D_{2^d}$ and showed that they have quantum symmetry. Finally, we have touched upon structures of quantum permutations, relationships between fractional and quantum isomorphisms as well as connection to quantum independence and chromatic numbers.

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