Two Problems on Bipartite Graphs

Erdos proved the well-known result that every graph has a spanning, bipartite subgraph such that every vertex has degree at least half of its original degree. Bollobas and Scott conjectured that one can get a slightly weaker result if we require the subgraph to be not only spanning and bipartite, but also balanced. We prove this conjecture for graphs of maximum degree 3. The majority of the paper however, will focus on graph tiling. Graph tiling (or sometimes referred to as graph packing) is where, given a graph H, we find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove an asymptotic minimum degree condition for an arbitrary bipartite graph G to be tiled by another arbitrary bipartite graph H. This proves a conjecture of Zhao and also implies an asymptotic version of a result of Kuhn and Osthus for bipartite graphs

07391 Abstracts Collection -- Probabilistic Methods in the Design and Analysis of Algorithms

From 23.09.2007 to 28.09.2007, the Dagstuhl Seminar 07391 "Probabilistic Methods in the Design and Analysis of Algorithms\u27\u27was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. The seminar brought together leading researchers in probabilistic methods to strengthen and foster collaborations among various areas of Theoretical Computer Science. The interaction between researchers using randomization in algorithm design and researchers studying known algorithms and heuristics in probabilistic models enhanced the research of both groups in developing new complexity frameworks and in obtaining new algorithmic results. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

An Extension of Ramsey\u27s Theorem to Multipartite Graphs

Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey\u27s Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number

Discrete Harmonic Analysis. Representations, Number Theory, Expanders and the Fourier Transform

This self-contained book introduces readers to discrete harmonic analysis with an emphasis on the Discrete Fourier Transform and the Fast Fourier Transform on finite groups and finite fields, as well as their noncommutative versions. It also features applications to number theory, graph theory, and representation theory of finite groups. Beginning with elementary material on algebra and number theory, the book then delves into advanced topics from the frontiers of current research, including spectral analysis of the DFT, spectral graph theory and expanders, representation theory of finite groups and multiplicity-free triples, Tao's uncertainty principle for cyclic groups, harmonic analysis on GL(2,Fq), and applications of the Heisenberg group to DFT and FFT. With numerous examples, figures, and over 160 exercises to aid understanding, this book will be a valuable reference for graduate students and researchers in mathematics, engineering, and computer science

Regularity and removal lemmas and their applications

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017.Cataloged from PDF version of thesis.Includes bibliographical references (pages 123-127).In this thesis, we analyze the regularity method pioneered by Szemerédi, and also discuss one of its prevalent applications, the removal lemma. First, we prove a new lower bound on the number of parts required in a version of Szemerédi's regularity lemma, determining the order of the tower height in that version up to a constant factor. This addresses a question of Gowers. Next, we turn to algorithms. We give a fast algorithmic Frieze-Kannan (weak) regularity lemma that improves on previous running times. We use this to give a substantially faster deterministic approximation algorithm for counting subgraphs. Previously, only an exponential dependence of the running time on the error parameter was known; we improve it to a polynomial dependence. We also revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for some co-NP-complete problems. We show how to use the Frieze-Kannan regularity lemma to approximate the regularity of a pair of vertex sets. We also show how to quickly find, for each [epsilon]' > [epsilon], an [epsilon]'-regular partition with k parts if there exists an [epsilon]-regular partition with k parts. After studying algorithms, we turn to the arithmetic setting. Green proved an arithmetic regularity lemma, and used it to prove an arithmetic removal lemma. The bounds obtained, however, were tower-type, and Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. The previous best known bound was tower-type with a logarithmic tower height. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in Fn/p. Finally, we give a new proof of a regularity lemma for permutations, improving the previous tower-type bound on the number of parts to an exponential bound.by László Miklós Lovász.Ph. D