14 research outputs found
Twin-width IV: ordered graphs and matrices
We establish a list of characterizations of bounded twin-width for
hereditary, totally ordered binary structures. This has several consequences.
First, it allows us to show that a (hereditary) class of matrices over a finite
alphabet either contains at least matrices of size , or at
most for some constant . This generalizes the celebrated Stanley-Wilf
conjecture/Marcus-Tardos theorem from permutation classes to any matrix class
over a finite alphabet, answers our small conjecture [SODA '21] in the case of
ordered graphs, and with more work, settles a question first asked by Balogh,
Bollob\'as, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes
of ordered graphs. Second, it gives a fixed-parameter approximation algorithm
for twin-width on ordered graphs. Third, it yields a full classification of
fixed-parameter tractable first-order model checking on hereditary classes of
ordered binary structures. Fourth, it provides a model-theoretic
characterization of classes with bounded twin-width.Comment: 53 pages, 18 figure
Small permutation classes
We establish a phase transition for permutation classes (downsets of
permutations under the permutation containment order): there is an algebraic
number , approximately 2.20557, for which there are only countably many
permutation classes of growth rate (Stanley-Wilf limit) less than but
uncountably many permutation classes of growth rate , answering a
question of Klazar. We go on to completely characterize the possible
sub- growth rates of permutation classes, answering a question of
Kaiser and Klazar. Central to our proofs are the concepts of generalized grid
classes (introduced herein), partial well-order, and atomicity (also known as
the joint embedding property)
Generating Permutations with Restricted Containers
We investigate a generalization of stacks that we call
-machines. We show how this viewpoint rapidly leads to functional
equations for the classes of permutations that -machines generate,
and how these systems of functional equations can frequently be solved by
either the kernel method or, much more easily, by guessing and checking.
General results about the rationality, algebraicity, and the existence of
Wilfian formulas for some classes generated by -machines are
given. We also draw attention to some relatively small permutation classes
which, although we can generate thousands of terms of their enumerations, seem
to not have D-finite generating functions
On The Growth Of Permutation Classes
We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order.
First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions m × 1 for some m, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape.
We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated graph and deduce some facts relating to the set of grid class growth rates. In the process, we establish a new result concerning tours on graphs. We also prove a similar result relating the growth rate of a geometric grid class to the matching polynomial of a graph, and determine the effect of edge subdivision on the matching polynomial. We characterise the growth rates of geometric grid classes in terms of the spectral radii of trees.
We then investigate the set of growth rates of permutation classes and establish a new upper bound on the value above which every real number is the growth rate of some permutation class. In the process, we prove new results concerning expansions of real numbers in non-integer bases in which the digits are drawn from sets of allowed values.
Finally, we introduce a new enumeration technique, based on associating a graph with each permutation, and determine the generating functions for some previously unenumerated classes. We conclude by using this approach to provide an improved lower bound on the growth rate of the class of permutations avoiding the pattern 1324. In the process, we prove that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution
The Epstein-Glaser approach to pQFT: graphs and Hopf algebras
The paper aims at investigating perturbative quantum field theory (pQFT) in
the approach of Epstein and Glaser (EG) and, in particular, its formulation in
the language of graphs and Hopf algebras (HAs). Various HAs are encountered,
each one associated with a special combination of physical concepts such as
normalization, localization, pseudo-unitarity, causality and an associated
regularization, and renormalization. The algebraic structures, representing the
perturbative expansion of the S-matrix, are imposed on the operator-valued
distributions which are equipped with appropriate graph indices. Translation
invariance ensures the algebras to be analytically well-defined and graded
total symmetry allows to formulate bialgebras. The algebraic results are given
embedded in the physical framework, which covers the two recent EG versions by
Fredenhagen and Scharf that differ with respect to the concrete recursive
implementation of causality. Besides, the ultraviolet divergences occuring in
Feynman's representation are mathematically reasoned. As a final result, the
change of the renormalization scheme in the EG framework is modeled via a HA
which can be seen as the EG-analog of Kreimer's HA.Comment: 52 pages, 5 figure
Disjointness of Linear Fractional Actions on Serre Trees
Serre showed that, for a discrete valuation field, the group of linear fractional transformations acts on an infinite regular tree with vertex degree determined by the residue degree of the field. Since the p-adics and the polynomials over the finite field of order p act on isomorphic trees, we may ask whether pairs of actions from these two groups are ever conjugate as tree automorphisms. We analyze permutations induced on finite vertex sets, and show a permutation classification result for actions by these linear fractional transformation groups. We prove that actions by specific subgroups of these groups are conjugate only in specific special cases
Substructure sensities in extremal combinatorics
One of the primary goals of extremal combinatorics is to understand how an object’s properties are influenced by the presence or multiplicity of a given substructure. Most classical theorems in the area, such as Mantel’s Theorem, are phrased in terms of substructure counts such as the number of edges or the number of triangles in a graph. Gradually, however, it has become more popular to express results in terms of the density of substructures, where the substructure counts are normalised by some natural quantity. This approach has several benefits; results are often more succinctly stated using densities, and it becomes easier to focus on the asymptotic behaviour of objects.
In this thesis, we study three topics concerning density. We begin Chapter 1 by contextualising the study of combinatorial density and justifying its importance within extremal combinatorics. We also introduce the relevant combinatorial objects, results, and questions that are central to the later chapters. Particular attention is paid to developing the theory of graph limits and flag algebras, two modern fields that rely heavily on the notion of density.
In Chapter 2, we investigate the interplay between the densities of cycles of length 3 and 4 in large tournaments. In particular, we prove two cases of a conjecture of Linial and Morgenstern (2016) that the minimum density of 4-cycles in a graph with a fixed density of 3-cycles is attained by a particular random construction.
In Chapter 3, we explore quasirandom permutations. A permutation is said to be quasirandom if the density of every subpermutation matches the expected density in a random permutation. Our main result is that quasirandomness can be characterised by a property which, on the surface, appears significantly weaker.
Lastly, in Chapter 4, we resolve a problem posed by Bubeck and Linial (2016) on the inducibility of trees. The inducibility of a tree X is defined as the maximum possible density of X in a large tree. We show that there exist non-path, non-star trees with positive inducibility, but that all such trees have inducibility bounded away from 1. We also show that there exists a sequence of trees in which every possible subtree appears asymptotically with positive density