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

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

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A Dynamic Survey of Graph Labeling

Extremal Graph Theory

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

Large Networks and Graph Limits

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values.

Small Ramsey Numbers

Let denote the class of all graphsG which satisfyG→(G
1,G
2). As a way of measuring minimality for members of
 , we define thesize Ramsey number ř(G
1,G
2) by
 . We then investigate various questions concerned with the asymptotic behaviour ofř.

The size ramsey number

All Ramsey numbers for cycles in graphs

An edge-coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge-coloring of a simple graph G is denoted by . A simple count shows that where ni denotes the number of vertices of degree i in G. We prove that where C is a constant depending only on Δ. Some results for special classes of graphs, notably trees, are also presented. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 73–82, 1997

Vertex-distinguishing proper edge-colorings

An adjacent vertex distinguishing edge-coloring of a simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors $\chi^\prime_a(G)$ required to give $G$ an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove $\chi^\prime_a(G)\le5$ for such graphs with maximum degree $\Delta(G)=3$ and prove $\chi^\prime_a(G)\le\Delta(G)+2$ for bipartite graphs. These bounds are tight. For $k$-chromatic graphs $G$ without isolated edges we prove a weaker result of the form $\chi^\prime_a(G)=\Delta(G)+O(\log k)$.

Adjacent Vertex Distinguishing Edge-Colorings

Problems and results are presented concerning the strong chromatic index, where the strong chromatic index is the smallest k such that the edges of the graph can be k-colored with the property that each color class is an induced matching. This parameter was suggested by Erdos and Nesetril and it is related to an extremal problem of Bermond, Bond and Peyrat concerning induced matchings of graphs.

Richard H. Schelp

Papers

The size ramsey number

All Ramsey numbers for cycles in graphs

Vertex-distinguishing proper edge-colorings

Adjacent Vertex Distinguishing Edge-Colorings

The strong chromatic index of graphs