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Cycles of even length in graphs
Abstract In this paper we solve a conjecture of P. Erdos by showing that if a graph G n has n vertices and at least 100kn 1+ 1 k edges, then G contains a cycle C 2 l of length 2 l for every integer lExpand
Random Walks in a Convex Body and an Improved Volume Algorithm
A randomized algorithm using O(n7 log’ n) separation calls to approximate the volume of a convex body with a fixed relative error is given and the mixing rate of Markov chains from finite to arbitrary Markov Chains is analyzed. Expand
Isoperimetric problems for convex bodies and a localization lemma
This lemma is a general “Localization Lemma” that reduces integral inequalities over then-dimensional space to integral inequalities in a single variable and is illustrated by showing how a number of well-known results can be proved using it. Expand
Szemeredi''s Regularity Lemma and its applications in graph theory
Szemer\''edi''s Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by random-looking graphs. Therefore the lemma helps inExpand
Supersaturated graphs and hypergraphs
We shall consider graphs (hypergraphs) without loops and multiple edges. Let ℒ be a family of so called prohibited graphs and ex (n, ℒ) denote the maximum number of edges (hyperedges) a graphExpand
Compactness results in extremal graph theory
The main purpose of this paper is to prove some compactness results for the case when L consists of cycles, and one of the main tools will be finding lower bounds on the number of pathsPk+1 in a graph ofn vertices andE edges. Expand
Random walks and an O*(n5) volume algorithm for convex bodies
This work introduces three new ideas: the use of the isotropic position (or at least an approximation of it) for rounding, the separation of global obstructions and local obstructions for fast mixing, and a stepwise interlacing of rounding and sampling. Expand
On a valence problem in extremal graph theory
Let L Kinp be a p-chromatic graph and e be an edge of L such that L - e is (p-1)-chromatic If G^n is a graph of n vertices without containing L but containing K"p, then the minimum valence of G^n is =
On the number of complete subgraphs of a graph II
Generalizing some results of P. Erdős and some of L. Moser and J. W. Moon we give lower bounds on the number of complete p-graphs K p of graphs in terms of the numbers of vertices and edges. Further,Expand
The History of Degenerate (Bipartite) Extremal Graph Problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe manyExpand