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

From social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.

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Introduction to random graphs

Given a family of (hyper)graphs $\mathcal{F}$ a (hyper)graph $G$ is said to be $\mathcal{F}$-saturated if $G$ is $F$-free for any $F \in\mathcal{F}$ but for any edge e in the complement of $G$ the (hyper)graph $G + e$ contains some $F\in\mathcal{F}$. We survey the problem of determining the minimum size of an $\mathcal{F}$-saturated (hyper)graph and collect many open problems and conjectures.

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A Survey of Minimum Saturated Graphs

We transpose the theory of rank metric. Gabidulin codes (8) W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes. Cambridge. Leech's construction is based on the binary Golay code of length. 24. Later on (13) V.Pless, An Introduction to the Theory of Error-Correcting Codes, Wi. Download full text in PDF Opens in a new window. R. Brualdi and V. Pless, On the length of codes with a given covering radius, Preprint. Introduction to the Theory of Functions of One VariableA. Characterization of (2(q + 1) + 2, 2, t, q)-min-hyper in PG(t, q) (t 3, q 5) and its applications to error-correcting codes. of Coding Theory. 5. 1.

http://ieeexplore.ieee.org/iel5/5/31339/01457228.pdf

Introduction to the theory of error-correcting codes

The 1-2-3 Conjecture, posed in 2004 by Karonski, Luczak, and Thomason, is as follows: "If G is a graph with no connected component having exactly 2 vertices, then the edges of G may be assigned weights from the set {1,2,3} so that, for any adjacent vertices u and v, the sum of weights of edges incident to u differs from the sum of weights of edges incident to v." This survey paper presents the current state of research on the 1-2-3 Conjecture and the many variants that have been proposed in its short but active history.

The 1-2-3 Conjecture and related problems: a survey

In the random hypergraph $H=H_{n,p;3}$ each possible triple appears independently with probability $p$. A loose Hamilton cycle can be described as a sequence of edges $\{x_i,y_i,x_{i+1}\}$ for $i=1,2,\ldots,n/2$ where $x_1,x_2,\ldots,x_{n/2},y_1,y_2,\ldots,y_{n/2}$ are all distinct. We prove that there exists an absolute constant $K>0$ such that if $p\geq {K\log n\over n^2}$ then $$\lim_{\textstyle{n\to \infty\atop 4|n}}\Pr(H_{n,p;3}\ contains\ a\ loose\ Hamilton\ cycle)=1.$$

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Loose Hamilton Cycles in Random Uniform Hypergraphs

The size-Ramsey number ${R}{F}$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any coloring of the edges of $G$ with two colors yields a monochromatic copy of $F$. In this paper, first we focus on the size-Ramsey number of a path $P_n$ on $n$ vertices. In particular, we show that $5n/2-15/2 \le {R}(){P_n} \le 74n$ for $n$ sufficiently large. (The upper bound uses expansion properties of random $d$-regular graphs.) This improves the previous lower bound, ${R}{P_n} \ge (1+\sqrt{2})n-O(1)$, due to Bollobas, and the upper bound, ${R}(){P_n} \le 91n$, due to Letzter. Next we study long monochromatic paths in an edge-colored random graph $\mathcal{G}(n,p)$ with $pn \to \infty$. Let $\alpha > 0$ be an arbitrarily small constant. Recently, Letzter showed that asymptotically almost surely (a.a.s.) any $2$-edge coloring of $\mathcal{G}(n,p)$ yields a monochromatic path of length $(2/3-\alpha)n$, which is optimal. Extending this result, we show that ...

On some Multicolor Ramsey Properties of Random Graphs

Given a graph G = (V; E) and a weight function omega : E -\textgreater R, a coloring of vertices of G, induced by omega, is defined by chi(omega) (nu) = Sigma(e(sic)nu) omega (e) for all nu is an element of V. In this paper, we show that determining whether a particular graph has a weighting of the edges from \1, 2\ that induces a proper vertex coloring is NP-complete.

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On the complexity of vertex-coloring edge-weightings

In this paper we show that 1=n is the asymptotic threshold for the existence of tight Hamilton cycles in random k-uniform hypergraphs, for all k � 3. We also determine thresholds for the existence of other types of Hamilton cycles.

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Andrzej Dudek

Papers

Loose Hamilton Cycles in Random Uniform Hypergraphs

On some Multicolor Ramsey Properties of Random Graphs

On the complexity of vertex-coloring edge-weightings

Tight Hamilton cycles in random uniform hypergraphs

An Alternative Proof of the Linearity of the Size-Ramsey Number of Paths