Asymptotic study of subcritical graph classes

International audienceWe present a unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp. $g_n$) of labelled (resp. unlabelled) graphs on $n$ vertices from a subcritical graph class {\cG}=\cup_n {\cG_n} satisfies asymptotically the universal behaviour $$g_n = c n^{-5/2} \gamma^n\ (1+o(1))$$ for computable constants $c,\gamma$, e.g. $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that the number of vertices of degree $k$ ($k$ fixed) in a graph chosen uniformly at random from \cG_n, converges (after rescaling) to a normal law as $n\to\infty$

Local Structure for Vertex-Minors

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''

The Mori fan of the Dolgachev–Nikulin–Voisin family in genus 2

In this paper we study the Mori fan of the Dolgachev–Nikulin–Voisin family in degree 2 as well as the associated secondary fan. The main result is an enumeration of all maximal dimensional cones of the two fans. © by the author(s)

Improving the capacity of radio spectrum: exploration of the acyclic orientations of a graph

The efficient use of radio spectrum depends upon frequency assignment within a telecommunications network. The solution space of the frequency assignment problem is best described by the acyclic orientations of the network. An acyclic orientation Ɵ of a graph (network) G is an orientation of the edges of the graph which does not create any directed cycles. We are primarily interested in how many ways this is possible for a given graph, which is the count of the number of acyclic orientations, a(G). This is just the evaluation of the chromatic polynomial of the graph χ(G; λ) at λ = -1. Calculating (and even approximating) the chromatic polynomial is known to be #P-hard, but it is unknown whether or not the approximation at the value -1 is. There are two key contributions in this thesis. Firstly, we obtain computational results for all graphs with up to 8 vertices. We use the data to make observations on the structure of minimal and maximal graphs, by which we mean graphs with the fewest and greatest number of acyclic orientations respectively, as well as on the distribution of acyclic orientations. Many conjectures on the structure of extremal graphs arise, of which we prove some in the theoretical part of the thesis. Secondly, we present a compression move which is monotonic with respect to the number of acyclic orientations, and with respect to various other parameters in particular cliques. This move gives us a new approach to classifying all minimal graphs. It also enables us to tackle the harder problem of identifying maximal graphs. We show that certain Turán graphs are uniquely maximal (Turán graphs are complete multipartite graphs with all vertex classes as equal as possible), and conjecture that all Turán graphs are maximal. In addition we derive an explicit formula for the number of acyclic orientations of complete bipartite graphs

Gonality of metric graphs and Catalan-many tropical morphisms to trees

This thesis consists of two points of view to regard degree-(g′+1) tropical morphisms Φ : (Γ,w) → Δ from a genus-(2g′) weighted metric graph (Γ,w) to a metric tree Δ, where g′ is a positive integer. The first point of view, developed in Part I, is purely combinatorial and constructive. It culminates with an application to bound the gonality of (Γ,w). The second point of view, developed in Part II, incorporates category theory to construct a unified framework under which both Φ and higher dimensional analogues can be understood. These higher dimensional analogues appear in the construction of a moduli space Gtrop/g→0,d parametrizing the tropical morphisms Φ, and a moduli spaceMtrop/g parametrizing the (Γ,w). There is a natural projection map Π : Gtrop/g→0,d →Mtrop/g that sends Φ : (Γ,w) → Δ to (Γ,w). The strikingly beautiful result is that when g = 2g′ and d = g′+1, the projection Π itself is an indexed branched cover, thus having the same nature as the maps Φ that are being parametrized. Moreover, fibres of Π have Catalan-many points. Each part has its own introduction that motivates and describes the problem from its own perspective. Part I and its introduction are based on two articles which are joint work with Jan Draisma. Part II contains material intended to be published as two articles. There is also a layman summary available at the beginning

On the Heegaard Floer homology of Dehn surgery and unknotting number

n this thesis we generalise three theorems from the literature on Heegaard Floer homology and Dehn surgery: one by Ozsv ́ath and Szab ́o on deficiency symmetries in half-integral L -space surgeries, and two by Greene which use Donaldson’s diagonali- sation theorem as an obstruction to integral and half-integral L -space surgeries. Our generalisation is two-fold: first, we eliminate the L -space conditions, opening these techniques up for use with much more general 3-manifolds, and second, we unify the integral and half-integral surgery results into a broader theorem applicable to non- zero rational surgeries in S 3 which bound sharp, simply connected, negative-definite smooth 4-manifolds. Such 3-manifolds are quite common and include, for example, a huge number of Seifert fibred spaces. Over the course of the first three chapters, we begin by introducing background material on knots in 3-manifolds, the intersection form of a simply connected 4- manifold, Spin- and Spin c -structures on 3- and 4-manifolds, and Heegaard Floer ho- mology (including knot Floer homology). While none of the results in these chapters are original, all of them are necessary to make sense of what follows. In Chapter 4, we introduce and prove our main theorems, using arguments that are predominantly algebraic or combinatorial in nature. We then apply these new theorems to the study of unknotting number in Chapter 5, making considerable headway into the extremely difficult problem of classifying the 3-strand pretzel knots with unknotting number one. Finally, in Chapter 6, we present further applications of the main theorems, ranging from a plan of attack on the famous Seifert fibred space realisation problem to more biologically motivated problems concerning rational tangle replacement. An appendix on the implications of our theorems for DNA topology is provided at the end.Open Acces

The Mori fan of the Dolgachev-Nikulin-Voisin family in genus 22

In this paper we study the Mori fan of the Dolgachev-Nikulin-Voisin family in degree $2$ as well as the associated secondary fan. The main result is an enumeration of all maximal dimensional cones of the two fans.Comment: 80 pages. Comments welcome

Combinatorics and Gauge-String Duality.

PhDThis thesis exhibits a range of applications of combinatoric methods to string theory. The concepts and techniques used in the counting of ribbon graphs, the theory of finite groups, and the construction of cell complexes can give powerful methods and interesting insights into the nature of gauge-string duality, the limits of CFT factorisation, and the topology of worldsheet moduli space. The first part presents a candidate space-time theory of the Belyi string with a holographic extension to three-dimensional Euclidean gravity. This is a model of gauge-string duality in which the correlators of the Gaussian Hermitian matrix model are identfied with sums over worldsheet embeddings onto the 2-sphere target space. We show that the matrix model can be reformulated on the sphere by using su(2) representation couplings, and that the analogues of Feynman diagrams in this model can be holographically extended to 3-manifolds within the Ponzano-Regge model. The second part explores the limits of large N factorisation in conformal field theory and the dual interpretation in supergravity. By considering exact finite N correlators of single and multi-trace half-BPS operators in N = 4 super Yang-Mills theory in four dimensions, we can explicitly nd the exact threshold of the operator dimensions at which the correlators fail to factorise. In the dual supergravity, this is the energy regime at which quantum correlations between distinct gravitons become non-vanishing. The third part develops a cell decomposition of the moduli space of punctured Riemann surfaces. The cells are specified by a particular family of ribbon graphs, and we show that these graphs correspond to equivalence classes of permutation tuples arising from branched coverings of the Riemann sphere. This description yields efficient computational approaches for understanding the topology of moduli spaceSEPNe