Finite Geometry and Extremal Graph Theory

On monomial graphs of girth eight

In this thesis, we focus on two problems in extremal graph theory. Extremal graph theory consists of all problems related to optimizing parameters defined on graphs. The concept of ``editing'' appears in many key results and techniques in extremal graph theory, either as a means to account for error in structural results, or as a quantity to minimize or maximize. A typical problem in spectral extremal graph theory seeks relationships between the extremes of certain graph parameters and the extremes of eigenvalues commonly associated to graphs. The \emph{edit distance problem} asks the following problem: for any fixed ``forbidden'' graph $F$, how many ``edits'' are needed to ensure that any graph on $n$ vertices can be made to contain no induced copies of $F$. If $F$ is a complete graph, then Tur\'{a}n's Theorem, an early fundamental result in extremal graph theory, provides a precise answer. The \emph{edit distance function} plays an essential role in answering this question and relates to the \emph{speed} of a graph hereditary property $\hh$ as well as the $\hh$-chromatic number of a random graph. The main techniques revolve around so-called \emph{colored regularity graphs (CRGs)}. We find an asymptotically almost sure formula for the edit distance function when $F$ is an Erd\H{o}s-R\'{e}nyi random graph whose density lies in $[1-1/\phi, 1/\phi]\approx [0.382¸0.618]$. As an intermediate step, we make several advances on the application of CRGs, such as the introduction and application of \emph{$p$-prohibited CRGs}. %In \emph{spectral graph theory}, we ask: given graph $G$ and some matrix $M$ which may be naturally associated to $G$, what do the eigenvalues of $M$ say about $G$? For any $n$-vertex graph $G$, its adjacency matrix $A = A_G$ is the $\{0,1\}$-valued $n\times n$ matrix whose $(u,v)$ entry indicates whether $uv$ is an edge of $G$. In $1999$, Gregory, Hershkowitz, and Kirkland defined the \emph{(adjacency) spread} of a graph as the difference between the maximum and minimum eigenvalues of its adjacency matrix. In their paper, since cited $68$ times, the authors conjectured that the graph on $n$ vertices which maximizes spread is the join of a complete graph on $\lfloor 2n/3\rfloor$ vertices with an independent set on $\lceil n/3\rceil$ vertices. We prove this claim for all $n$ sufficiently large. As an intermediate step, we prove an analogous result for the eigenvalues of \emph{graphons} (equivalently, kernel operators on symmetric functions $W:[0,1]^2\to [0,1]$).

Proof of a conjecture on monomial graphs

In this chapter we concentrate on new applications of finite fields which emerged quite recently. Such applications include, but are not limited to, cryptography, extremal graph theory, polynomial mappings and complexity theory which are extremely quickly developing areas posing a number exciting questions directly related to the theory of finite fields. Many classical applications of finite fields to such areas of discrete mathematics as combinatorics (in the construction of a number of combinatorial designs), finite geometries and to other areas of discrete mathematics have been described in [279, 280, 445, 1258, 1741, 1743, 1808, 2816], see also Chapters 5, 6, and 7 of this book. Here we only discuss several works, which have to do with new applications of finite fields.

In this dissertation, we consider a wide range of problems in algebraic and extremal graph theory. In extremal graph theory, we will prove that the Tree Packing Conjecture is true for all sequences of trees that are 'almost stars'; and we prove that the Erdos-Sos conjecture is true for all graphs G with girth at least 5. We also conjecture that every graph G with minimal degree k and girth at least $2t+1$ contains every tree T of order $kt+1$ such that $\Delta(T)\leq k.$ This conjecture is trivially true for t = 1. We Prove the conjecture is true for t = 2 and that, for this value of t, the conjecture is best possible. We also provide supporting evidence for the conjecture for all other values of t. In algebraic graph theory, we are primarily concerned with isomorphism problems for vertex-transitive graphs, and with calculating automorphism groups of vertex-transitive graphs. We extend Babai's characterization of the Cayley Isomorphism property for Cayley hypergraphs to non-Cayley hypergraphs, and then use this characterization to solve the isomorphism problem for every vertex-transitive graph of order pq, where p and q distinct primes. We also determine the automorphism groups of metacirculant graphs of order pq that are not circulant, allowing us to determine the nonabelian groups of order pq that are Burnside groups. Additionally, we generalize a classical result of Burnside stating that every transitive group G of prime degree p, is doubly transitive or contains a normal Sylow p-subgroup to all $p\sp k,$ provided that the Sylow p-subgroup of G is one of a specified family. We believe that this result is the most significant contained in this dissertation. As a corollary of this result, one easily gives a new proof of Klin and Poschel's result characterizing the automorphism groups of circulant graphs of order $p\sp k,$ where p is an odd prime.

Extremal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:math>-free bipartite graphs

In recent years, characterizing the extremal (maximal or minimal) graphs in a given set of graphs with respect to some distance-based topological index has become an important direction in chemical graph theory.

A point in a finite projective plane PG(2, pn), may be denoted by the symbol (Xl, X2, X3), where the coordinates x1, X2, X3 are marks of a Galois field of order pn, GF(pn). The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (X1, X2, X3) and (kxl, kx2, kx3) are to be thought of as the same point. The totality of points whose coordinates satisfy the equation ulxl+u2x2+U3x3 = 0, where u1, U2, u3 are marks of the GF(pn), not all zero, is called a line. The plane then consists of p2n +pn + 1 = q points and q lines; each line contains pn+1 points.t A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian; it exists for every prime p and positive integer n, and there is only one such PG(2, pn) for a given p and n (VB, p. 247, VY, p. 151). Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane. (A collineation is a 1-1 transformation carrying points into points and lines into lines.) Suppose C carries Ao into Al, A1 into A2,... , Ak into Ao; or, denoting the product C C by C2, C. C2 by C3, etc., we have C(Ao) =A1, C2(Ao) =A2, . . , Ck(A o) =A o. If k is the smallest positive integer for which C k(A o) =Ao, we call k the period of C with respect to the point A o. If the period of a collineation C with respect to a point Ao is q (=p2n+pn+l), then the period of C with respect to any point in the plane is q, and in this case we will call C simply a collineation of period q. We prove in the first theorem that there is always at least one collineation of period q, and from it we derive some results of interest in finite geometry and number theory. Let

This conference was one of a series of Oberwolfach conferences on the same topic, held every two years. There were 55 participants, including about twenty graduate students and postdocs. Since graph theory is a broad and many-faceted field, we need to focus the workshop on a specific domain within the field (as we did in previous years). A dominant area within graph theory today is extremal graph theory: the study of the asymptotic and probabilistic behaviour of various graph parameters. While this is an exciting area, it is of a different character from much of the remainder of graph theory, and its inclusion would lead to an undesired division of the participants. Since extremal graph theory is adequately covered by the Oberwolfach combinatorics conference, we decided to minimize the extremal content and to focus the conference on other fundamental areas in graph theory, their interaction, and their interaction with mathematics outside combinatorics. In particular, we focus on structural aspects of graphs like decomposability, embeddability, duality, and noncontainment of substructures and its relations to basic questions like colourability and connectivity, and on the applicability of methods from algebra, geometry, and topology to these areas. The conference was organized along lines similar to the earlier Oberwolfach Graph Theory conferences of 2003 and 2005. There is a reduced number of formal talks, to give space for informal workshops on various topics in the area. As before, on the first day, we asked everyone to make a five-minute presentation of their current interests. This was designed also to promote contact between participants early in the meeting, and turns out to work very well. As for the talks intended for all the participants, there were six 50-minute and twenty-one 25-minute talks. We selected these from the abstracts submitted before the meeting, and we chose them to be of scientific relevance and general interest, as best we could. Also, we deliberately chose younger speakers, and tried for a wide range of topics. Among the highlights of the week were the presentation of a proof of Berge's strong path partition conjecture for k=2 , a new method to apply chip-firing games in graphs to derive a Riemann–Roch theorem in tropical geometry, and constructions of limits of graphs forming a topological space that encompass Szemeredi's regularity lemma. The workshops are intended to be informal – no formal speakers or time slots, with a ‘convenor’ to manage the workshop – and are focused on areas with specific recent results, conjectures and problems. It is our experience that such workshops can be the best part of a conference. We wanted these to be really informal, so that anyone in the group who wanted to contribute could spontaneously get up and say his or her piece. Before the meeting, we selected a few topics that seemed appropriate for workshops, some of which were suggested by participants. This time there were workshops on graph width, matroids, graph limits, flows and cycles in infinite graphs, and paths and minors. Sometimes, the workshops were scheduled in parallel, and some were extended in evenings later in the week. We were very satisfied with the way the conference worked out. Although the participants had varied interests, they were not so far apart that they polarized into separate camps. Most of the talks were of interest to almost all of of the participants, and the workshop format satisfied the desire of some for more focus. Again, we regard this as a very successful conference. If we organize another Graph Theory meeting at Oberwolfach, we would run it on the same lines. We are very thankful to the Oberwolfach management and staff for the opportunity to organize this meeting and for their smooth running and support of the meeting.

In this dissertation, we treat several problems in Ramsey theory, probabilistic combinatorics and extremal graph theory. We begin with the Ramsey theoretic problem of finding exactly m-coloured graphs. For which natural numbers m ∈ N are we guaranteed to find an m-coloured complete subgraph in any edge colouring of the complete graph on N? We resolve this question completely and prove, answering a question of Stacey and Weidl [104], that whenever we colour N(2) with infinitely many colours, we are guaranteed to find an ( n 2 ) -coloured complete subgraph for each n ∈ N. In addition, we also demonstrate that given a colouring of N(2) with k colours, there are at least √ 2k distinct values m ∈ [k] for which an infinite m-coloured complete subgraph exists. Finally, we also prove that given a colouring of N(2) with k colours and m ∈ [k], we can always find an infinite m-coloured complete subgraph for some m ∈ [k] such that |m− m| ≤ √ m/2. Next, we give some results in probabilistic combinatorics. First, we investigate the stability of the Erdős–Ko–Rado Theorem. For natural numbers n, r ∈ N with n ≥ r, the Kneser graph K(n, r) is the graph on the family of r-element subsets of {1, . . . , n} in which two sets are adjacent if and only if they are disjoint. Delete the edges of K(n, r) with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as r/n is bounded away from 1/2, even when the probability of retaining an edge of the Kneser graph is quite small; we also prove a much more precise result when r = o(n1/3). We then study a geometric bootstrap percolation model on the three dimensional grid [n]3 called line percolation. In line percolation with infection parameter r, infection spreads from a subset A ⊂ [n]3 of initially infected lattice points as follows: if there is an axis parallel line L with r or more infected lattice points on it, then every lattice point of [n]3 on L gets infected and we repeat this until the infection can no longer spread. Our main result is the determination the critical density of initially infected points at which percolation (infection of the entire grid) becomes likely. Finally, we present two results in extremal graph theory. First, we consider a graph partitioning problem. For a graph G, let f(G) be the largest integer k such that there are two vertex-disjoint subgraphs of G, each on k vertices, inducing the same number of edges. We prove that f(G) ≥ n/2 − o(n) for every graph G on n vertices, settling a conjecture of Caro and Yuster [36]. Finally, we study the problem of cops and robbers on the grid where the robber is allowed to move faster than the cops. We prove that when the speed of the robber is a sufficiently large constant, the number of cops needed to catch the robber on an n×n grid is exp(Ω(log n/ log log n)).

In recent years several classical results in extremal graph theory have been improved in a uniform way and their proofs have been simplified and streamlined. These results include a new Erdős-Stone-Bollobas theorem, several stability theorems, several saturation results and bounds for the number of graphs with large forbidden subgraphs. Another recent trend is the expansion of spectral extremal graph theory, in which extremal properties of graphs are studied by means of eigenvalues of various matrices. One particular achievement in this area is the casting of the central results above in spectral terms, often with additional enhancement. In addition, new, specific spectral results were found that have no conventional analogs. All of the above material is scattered throughout various journals, and since it may be of some interest, the purpose of this survey is to present the best of these results in a uniform, structured setting, together with some discussions of the underpinning ideas. Introduction The purpose of this survey is to give a systematic account of two recent lines of research in extremal graph theory. The first one, developed in [14],[15],[16],[63, 68], improves a number of classical results grouped around the theorem of Turan. The main progress is along the following three guidelines: replacing fixed parameters by variable ones; giving explicit conditions for the validity of the statements; developing and using tools of general scope.

We present results of the recent research on sparse graphs and finite structures in the context of of contemporary combinatorics, graph theory, model theory and mathematical logic, complexity of algorithms and probability theory. The topics include: complexity of subgraph- and homomorphism- problems; model checking problems for first order formulas in special classes; property testing in sparse classes of structures. All these problems can be studied under the umbrella of classes of structures which are Nowhere Dense and in the context of Nowhere Dense -- Somewhere Dense dichotomy. This dichotomy presents the classification of the general classes of structures which proves to be very robust and stable as it can be defined alternatively by most combinatorial extremal invariants as well as by algorithmic and logical terms. We give examples from logic, geometry and extremal graph theory. Finally we characterize the existence of all restricted dualities in terms of limit objects defined on the homomorphism order of graphs.

Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as property testing in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization). This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future. --Persi Diaconis, Stanford University This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great expositor. --Noga Alon, Tel Aviv University, Israel Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory. --Terence Tao, University of California, Los Angeles, CA Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovasz's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike. --Bela Bollobas, Cambridge University, UK

The [n,k,r]-Locally recoverable codes (LRC) studied in this work are a well-studied family of [n,k] linear codes for which the value of each symbol can be recovered by a linear combination of at most r other symbols. In this paper, we study the LMD problem, which is to find the largest possible minimum distance of [n,k,r]-LRCs, denoted by D(n,k,r). LMD can be approximated within an additive term of one—it is known that D(n,k,r) is equal to either d⁎ or d⁎−1, where d⁎=n−k−⌈kr⌉+2. Moreover, for a range of parameters, it is known precisely whether the distance D(n,k,r) is d⁎ or d⁎−1. However, the problem is still open despite a significant effort. In this work, we convert LMD to an equivalent simply-stated problem in graph theory. Using this conversion, we show that an instance of LMD is at least as hard as computing the size of a maximal graph of high girth, a hard problem in extremal graph theory. This is an evidence that LMD—although can be approximated within an additive term of one—is hard to solve in general. As a positive result, thanks to the conversion and the exiting results in extremal graph theory, we solve LMD for a range of code parameters that has not been solved before.

Extremal graph theory has a great number of conjectures concerning the embedding of large sparse graphs into dense graphs. Szemerdi's Regularity Lemma is a valuable tool in finding embeddings of sm...