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

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

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Graph Coloring Problems

This paper starts with the project of finding a large subclass of NP which exhibits a dichotomy. The approach is to find this subclass via syntactic prescriptions. While the paper does not achieve this goal, it does isolate a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit this dichotomy. We justify the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three restrictions do not show this dichotomy. We then explore the structure of this class. We show that all problems in this class reduce to the seemingly simpler class CSP. We divide CSP into subclasses and try to unify the collection of all known polytime algorithms for CSP problems and extract properties that make CSP problems NP-hard. This is where the second part of the title, "a study through Datalog and group theory," comes in. We present conjectures about this class which would end in showing the dichotomy.

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The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory

Circular chromatic number: a survey

The Game Coloring Number of Planar Graphs

For a graph $G$, let $\diam(G)$ denote the diameter of $G$. For any two vertices $u$ and $v$ in $G$, let $d(u, v)$ denote the distance between $u$ and $v$. A multilevel distance labeling (or distance labeling) for $G$ is a function $f$ that assigns to each vertex of $G$ a nonnegative integer such that for any vertices $u$ and $v$, $|f(u)-f(v)| \geq \diam(G) - d_G(u, v) +1$. The span of $f$ is the largest number in $f(V)$. The radio number of $G$, denoted by $rn(G)$, is the minimum span of a distance labeling for $G$. In this paper, we completely determine the radio numbers for paths and cycles.

Multilevel Distance Labelings for Paths and Cycles

Colouring graphs with bounded generalized colouring number

The oriented chromatic number o( ~ G) of an oriented graph ~ G =( V;A) is the minimum number of vertices in an oriented graph ~ H for which there exists a homomorphism of ~ G to ~ H. The oriented chromatic number o(G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relations between the oriented chromatic number and the acyclic chromatic number and some other parameters of a graph. We shall give a lower bound for o(G) in terms of a(G). An upper bound for o(G)in terms of a(G) was given by Raspaud and Sopena. We also give an upper bound for o(G) in terms of the maximum degree of G. We shall show that this upper bound is not far from being optimal. c 1997 John Wiley & Sons, Inc.

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Xuding Zhu

Papers

Circular chromatic number: a survey

The Game Coloring Number of Planar Graphs

Multilevel Distance Labelings for Paths and Cycles

Colouring graphs with bounded generalized colouring number

Acyclic and oriented chromatic numbers of graphs