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Friday, Dec 3, 2010 |
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Asymptotic enumeration of sparse 2-connected graphs and strongly connected digraphs |
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I will discuss an asymptotic formula for the number of 2-connected graphs on $n$ vertices and $m$ edges, provided that $m-n\to\infty$ and $m=O(n\log n)$ as $n\to\infty$. This is the entire range of $m$ not covered by previous results, solving a problem of Wright from 1983 and also determining his mysterious constant $a$. The proof uses properties of the core and kernel of random graphs with minimum degree at least 2. This is joint work with Graeme Kemkes and Cristiane Sato. For strongly connected digraphs, a similar gap has been filled, jointly with Xavier Perez-Gimenez. We use a directed analogue of the kernel. |