Hamiltonicity in multitriangular graphs

The family of 5-valent polyhedral graphs whose faces are all triangles or 3s-gons, s ≥ 9, is shown to contain non-hamiltonian graphs and to have a shortness exponent smaller than one

Optimal acyclic edge colouring of grid like graphs

AbstractWe determine the values of the acyclic chromatic index of a class of graphs referred to as d-dimensional partial tori. These are graphs which can be expressed as the cartesian product of d graphs each of which is an induced path or cycle. This class includes some known classes of graphs like d-dimensional meshes, hypercubes, tori, etc. Our estimates are exact except when the graph is a product of a path and a number of odd cycles, in which case the estimates differ by an additive factor of at most 1. Our results are also constructive and provide an optimal (or almost optimal) acyclic edge colouring in polynomial time

Hypohamiltonian and almost hypohamiltonian graphs

This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results

Probabilistic methods and coloring problems in graphs

Aquest projecte està dedicat a estudiar el k-èssim nombre cromàtic generalitzat que sorgeix de les descomposicions Low Tree--Depth en grafs usant mètodes probabilístics.. Una extensió natural del nombre cromàtic d'un graf és l'estudi de particions de grafs en les que cada i parts indueixen un subgraf amb un cert paràmetre acotat en funció de i, per exemple cada i parts tenen com a molt i-1 arestes. En particular el nombre cromàtic generalitzat és le mínim nombre de parts per tal que cada i parts té 'treedepth' com a molt i. Resultats recents proven que grans classes de grafs tenen paràmetres d'aquest tipus acotats. L'objectiu del projecte és (i) fer servie mètodes probabilístics per donar cotas ajustades d'aquests paràmetres i (ii) estudiar el seu valor per grafs aleatoris

Checkerboard CFT

The Checkerboard conformal field theory is an interesting representative of a large class of non-unitary, logarithmic Fishnet CFTs (FCFT) in arbitrary dimension which have been intensively studied in the last years. Its planar Feynman graphs have the structure of a regular square lattice with checkerboard colouring. Such graphs are integrable since each coloured cell of the lattice is equal to an R-matrix in the principal series representations of the conformal group. We compute perturbatively and numerically the anomalous dimension of the shortest single-trace operator in two reductions of the Checkerboard CFT: the first one corresponds to the fishnet limit of the twisted ABJM theory in 3D, whereas the spectrum in the second, 2D reduction contains the energy of the BFKL Pomeron. We derive an analytic expression for the Checkerboard analogues of Basso--Dixon 4-point functions, as well as for the class of Diamond-type 4-point graphs with disc topology. The properties of the latter are studied in terms of OPE for operators with open indices. We prove that the spectrum of the theory receives corrections only at even orders in the loop expansion and we conjecture such a modification of Checkerboard CFT where quantum corrections occur only with a given periodicity in the loop order.Comment: 66 pages, 24 figure

Renormalization in tensor field theory and the melonic fixed point

This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with $O(N)^3$ symmetry and long-range propagator. The existence of a non-perturbative fixed point in any $d$ at large $N$ is established. We found four lines of fixed points parametrized by the so-called tetrahedral coupling. One of them is infrared attractive, strongly interacting and gives rise to a new kind of CFT, called melonic CFTs which are then studied in more details. We first compute dimensions of bilinears and OPE coefficients at the fixed point which are consistent with a unitary CFT at large $N$. We then compute $1/N$ corrections. At next-to-leading order, the line of fixed points collapses to one fixed point. However, the corrections are complex and unitarity is broken at NLO. Finally, we show that this model respects the $F$-theorem. The next part of the thesis investigates sextic tensor field theories in rank $3$ and $5$. In rank $3$, we found two IR stable real fixed points in short range and a line of IR stable real fixed points in long range. Surprisingly, the only fixed point in rank $5$ is the Gaussian one. For the rank $3$ model, in the short-range case, we still find two IR stable fixed points at NLO. However, in the long-range case, the corrections to the fixed points are non-perturbative and hence unreliable: we found no precursor of the large $N$ fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large $N$ limit. We prove that models with tensors in an irreducible representation of $O(N)$ or $Sp(N)$ in rank $5$ indeed admit a large $N$ limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.Comment: PhD thesis, 277 pages. Based on papers: arXiv:1903.03578, arXiv:1909.07767, arXiv:1912.06641, arXiv:2007.04603, arXiv:2011.11276, arXiv:2104.03665, arXiv:2109.08034, arXiv:2111.1179

Problems of optimal choice on posets and generalizations of acyclic colourings

NOTE : The mathematical symbols in the abstract do not always display correctly in this text field. Please see the abstract in the thesis for the definitive abstract. ABSTRACT: This dissertation is in two parts, each of three chapters. In Part 1, I shall prove some results concerning variants of the `secretary problem'. In Part 2, I shall bound several generalizations of the acyclic chromatic number of a graph as functions of its maximum degree. I shall begin Chapter 1 by describing the classical secretary problem, in which the aim is to select the best candidate for the post of a secretary, and its solution. I shall then summarize some of its many generalizations that have been studied up to now, provide some basic theory, and briefly outline the results that I shall prove. In Chapter 2, I shall suppose that the candidates come as ‘m’ pairs of equally qualified identical twins. I shall describe an optimal strategy, a formula for its probability of success and the asymptotic behaviour of this strategy and its probability of success as m → ∞. I shall also find an optimal strategy and its probability of success for the analagous version with ‘c’-tuplets. I shall move away from known posets in Chapter 3, assuming instead that the candidates come from a poset about which the only information known is its size and number of maximal elements. I shall show that, given this information, there is an algorithm that is successful with probability at least ¹/e . For posets with ‘k ≥ 2’ maximal elements, I shall prove that if their width is also ‘k’ then this can be improved to ‘k-1√1/k’ and show that no better bound of this type is possible. In Chapter 4, I shall describe the history of acyclic colourings, in which a graph must be properly coloured with no two-coloured cycle, and state some results known about them and their variants. In particular, I shall highlight a result of Alon, McDiarmid and Reed, which bounds the acyclic chromatic number of a graph by a function of its maximum degree. My results in the next two chapters are of this form. I shall consider two natural generalizations in Chapter 5. In the first, only cycles of length at least ’l’ must receive at least three colours. In the second, every cycle must receive at least ‘c’ colours, except those of length less than ‘c’, which must be multicoloured. My results in Chapter 6 generalize the concept of a cycle; it is now subgraphs with minimum degree ‘r’ that must receive at least three colours, rather than subgraphs with minimum degree two (which contain cycles). I shall also consider a natural version of this problem for hypergraphs