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The structure of a random graph at the point of the phase transition. (English) Zbl 0807.05065

Standard random graph models \(G(n,M)\) and \(G(n,p)\) are considered with \(2M/n= np= 1+ \lambda n^{-1/3}\) where \(\lambda\) is a constant and \(n\to \infty\). A formula is given for the limiting distribution of the numbers of components having more edges than vertices. Limiting planarity probabilities are also given.

MSC:

05C80 Random graphs (graph-theoretic aspects)
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
05C30 Enumeration in graph theory
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