Modern Graph Theory

Proofs from THE BOOK, by Martin Aigner and Giinter M. Ziegler. Pp.199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag). I would like to begin with the authors' words from the Preface of this book: Paul Erd6s liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. ErdSs also said that 'you need not believe in God but, as a mathematician, you should believe in The Book'. This excellent book is dedicated to the memory of Paul ErdSs. Paul himself wrote several pages but he is not listed as a co-author because of his death in the summer of 1996. The idea of the book is to present beautiful proofs of some of the most exciting and historically important mathematical theorems. The book focuses on human endeavour to make the proofs more and more simple and striking.

Proofs from THE BOOK , by Martin Aigner and Günter M. Ziegler. Pp. 199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag).

Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to \emph{negative correlation} properties. However, what if an application naturally calls for dependent rounding on the one hand, and desires \emph{positive} correlation on the other? More generally, we develop algorithms that guarantee the known properties of dependent rounding, but also have nearly best-possible behavior - near-independence, which generalizes positive correlation - on "small" subsets of the variables. The recent breakthrough of Li & Svensson for the classical $k$-median problem has to handle positive correlation in certain dependent-rounding settings, and does so implicitly. We improve upon Li-Svensson's approximation ratio for $k$-median from $2.732 + \epsilon$ to $2.675 + \epsilon$ by developing an algorithm that improves upon various aspects of their work. Our dependent-rounding approach helps us improve the dependence of the runtime on the parameter $\epsilon$ from Li-Svensson's $N^{O(1/\epsilon^2)}$ to $N^{O((1/\epsilon) \log(1/\epsilon))}$.

An Improved Approximation for $k$-median, and Positive Correlation in Budgeted Optimization

This in-depth coverage of important areas of graph theory maintains a focus on symmetry properties of graphs. Standard topics on graph automorphisms are presented early on, while in later chapters more specialised topics are tackled, such as graphical regular representations and pseudosimilarity. The final four chapters are devoted to the reconstruction problem, and here special emphasis is given to those results that involve the symmetry of graphs, many of which are not to be found in other books. This second edition expands on several of the topics found in the first edition and includes both an enriched bibliography and a wide collection of exercises. Clearer proofs are provided, as are new examples of graphs with interesting symmetry properties. Any student who masters the contents of this book will be well prepared for current research in many aspects of the theory of graph automorphisms and the reconstruction problem.

Topics in Graph Automorphisms and Reconstruction

We give an efficient deterministic algorithm that extracts $\Omega(n^{2\gamma})$ almost-random bits from sources where $n^{\frac{1}{2}+\gamma}$ of the $n$ bits are uniformly random and the rest are fixed in advance. This improves upon previous constructions, which required that at least $n/2$ of the bits be random in order to extract many bits. Our construction also has applications in exposure-resilient cryptography, giving explicit adaptive exposure-resilient functions and, in turn, adaptive all-or-nothing transforms. For sources where instead of bits the values are chosen from $[d]$, for $d>2$, we give an algorithm that extracts a constant fraction of the randomness. We also give bounds on extracting randomness for sources where the fixed bits can depend on the random bits.

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Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography

The maximum number of complete subgraphs in a graph with given maximum degree

Defect Sauer results

Extending results of Christie and Irving, we examine the action of reversals and transpositions on finite strings over an alphabet of size k. We show that determining reversal, transposition, or signed reversal distance between two strings over a finite alphabet is NP-hard, while for "dense" instances we give a polynomial-time approximation scheme. We also give a number of extremal results, as well as investigating the distance between random strings and the problem of sorting a string over a finite alphabet.

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Reversals and Transpositions Over Finite Alphabets

The study of the number of independent sets in a graph has a rich history. Recently, Kahn proved that disjoint unions of $K_{r,r}$'s have the maximum number of independent sets amongst $r$-regular bipartite graphs. Zhao extended this to all $r$-regular graphs. If we instead restrict the class of graphs to those on a fixed number of vertices and edges, then the Kruskal-Katona theorem implies that the graph with the maximum number of independent sets is the lex graph, where edges form an initial segment of the lexicographic ordering. In this paper, we study three related questions. Firstly, we prove that the lex graph has the maximum number of weighted independent sets for any appropriate weighting. Secondly, we solve the problem of maximizing the number of independents sets in graphs with specified independence number or clique number. Finally, for $m\leq n$, we find the graphs with the minimum number of independent sets for graphs with $n$ vertices and $m$ edges.

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Extremal problems for independent set enumeration

The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of homomorphisms into K2 with one looped vertex (or equivalently, the largest number of independent sets) among graphs with fixed number of vertices and edges. Our main result is the solution to the extremal problem for the number of homomorphisms into P**math-image**, the completely looped path of length 2 (known as the Widom–Rowlinson model in statistical physics). We show that there are extremal graphs that are threshold, give explicitly a list of five threshold graphs from which any threshold extremal graph must come, and show that each of these “potentially extremal” threshold graphs is in fact extremal for some number of edges. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 261–284, 2011 © 2011 Wiley Periodicals, Inc.

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A. J. Radcliffe

Papers

The maximum number of complete subgraphs in a graph with given maximum degree

Defect Sauer results

Reversals and Transpositions Over Finite Alphabets

Extremal problems for independent set enumeration

Extremal graphs for homomorphisms