An undirected graph on n vertices is disconnected if there...

An undirected graph on $n$ vertices is disconnected if there exists a set of $k<n$ vertices such that there is no edge between this set and the rest of the graph. Otherwise, the graph is said to be connected. Show that there exists a constant $c$ such that if $N \geq c n \log n$ then, with probability $O\left(\mathrm{e}^{-n}\right)$, a graph randomly chosen from $G_{n, N}$ is connected.

Key Concepts

Probability Bounds for Connectivity

Probability bounds for connectivity in random graphs quantify how likely it is that a graph exhibits the connected property given a certain number of edges. Specifically, the result indicates that if the graph has at least c n log n edges (for an appropriate constant c), then the probability that the graph remains disconnected is exponentially small, on the order of e^{-n}. This bound demonstrates the robustness of connectivity in complex networks as the size increases.

Graph Connectivity

Graph connectivity is the property of an undirected graph where there is a path between every pair of vertices. This means that the graph is entirely 'one piece', with no isolated subgraphs or components. The study of connectivity is central in graph theory and has implications in network theory, communication, and many areas of combinatorics.

Disconnected Graph

A disconnected graph is one where there exists at least one partition of the vertex set into two nonempty subsets such that there is no edge linking any vertex from one subset to any vertex from the other. This concept is crucial for understanding the structure and resilience of networks and forms the basis for arguments in proofs involving connectivity thresholds.

Threshold Phenomenon

The threshold phenomenon refers to the sudden change in the property of random graphs when a parameter (often the number of edges) crosses a critical value. In the context of graph connectivity, it is observed that when the number of edges is of the order of n log n, the graph transitions from being likely disconnected to being likely connected. This concept is pivotal in determining the minimal conditions required for almost sure connectivity.

Random Graph Model

The random graph model, such as G(n, N), involves constructing graphs by randomly selecting a fixed number of edges among a specified number of vertices. This model is used to study the probabilistic behavior of graph properties, such as connectivity, as the number of edges varies. It provides a framework to understand typical behavior of large networks.

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