Łuczak, Tomasz; Pittel, Boris; Wierman, John C. The structure of a random graph at the point of the phase transition. (English) Zbl 0807.05065 Trans. Am. Math. Soc. 341, No. 2, 721-748 (1994). 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. Reviewer: O.Frank (Stockholm) Cited in 4 ReviewsCited in 52 Documents 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 Keywords:phase transition; planar graph; threshold function; random graph; limiting distribution; planarity probabilities PDFBibTeX XMLCite \textit{T. Łuczak} et al., Trans. Am. Math. Soc. 341, No. 2, 721--748 (1994; Zbl 0807.05065) Full Text: DOI