We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

Introduction to coding theory (2nd edition), by J.H. van Lint. Pp.183 DM86. 1992. ISBN 0-387-54894-7 (Springer)

In a recent breakthrough, Keevash proved the Existence conjecture for combinatorial designs, which has its roots in the 19th century. We give a new proof, based on the method of iterative absorption. Our main result concerns $K^{(r)}_{q}$-decompositions of hypergraphs whose clique distribution fulfils certain uniformity criteria. These criteria offer considerable flexibility. This enables us to strengthen the results of Keevash as well as to derive a number of new results, for example a resilience version and minimum degree version.

The existence of designs via iterative absorption

This pedagogical and self-contained text describes the modern mean field theory of simple structural glasses. The book begins with a thorough explanation of infinite-dimensional models in statistical physics, before reviewing the key elements of the thermodynamic theory of liquids and the dynamical properties of liquids and glasses. The central feature of the mean field theory of disordered systems, the existence of a large multiplicity of metastable states, is then introduced. The replica method is then covered, before the final chapters describe important, advanced topics such as Gardner transitions, complexity, packing spheres in large dimensions, the jamming transition, and the rheology of glass. Presenting the theory in a clear and pedagogical style, this is an excellent resource for researchers and graduate students working in condensed matter physics and statistical mechanics.

Theory of Simple Glasses: Exact Solutions in Infinite Dimensions

We prove χ′ s ( G ) ≤ 1.93 Δ( G ) 2 for graphs of sufficiently large maximum degree where χ′ s ( G ) is the strong chromatic index of G . This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where we are allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.

A Stronger Bound for the Strong Chromatic Index

https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.081/Henning/strchrom.pdf

A stronger bound for the strong chromatic index

We prove that a cubic graph with $m$ edges has an induced matching with at least $m/9$ edges. Our result generalizes a result for planar graphs due to Kang, Mnich, and Muller (SIAM J. Discrete Math., 26 (2012), pp. 1383--1411) and solves a conjecture of Henning and Rautenbach.

Induced Matchings in Subcubic Graphs

We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gyarfas and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first $o(n)$ trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemeredi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.

Felix Joos

Papers

A Stronger Bound for the Strong Chromatic Index

A stronger bound for the strong chromatic index

Induced Matchings in Subcubic Graphs

Optimal packings of bounded degree trees

On kissing numbers and spherical codes in high dimensions