Abstract
We study various properties of a random graph R n , drawn uniformly at random from the class \( \mathcal{A}_n \) of all simple graphs on n labelled vertices that satisfy some given property, such as being planar or having tree-width at most κ. In particular, we show that if the class \( \mathcal{A} \) is’ small’ and ‘addable’, then the probability that R n is connected is bounded away from 0 and from 1. As well as connectivity we study the appearances of subgraphs, and thus also vertex degrees and the numbers of automorphisms. We see further that if \( \mathcal{A} \) is’ smooth’ then we can make much more precise statements for example concerning connectivity.
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McDiarmid, C., Steger, A., Welsh, D.J.A. (2006). Random Graphs from Planar and Other Addable Classes. In: Klazar, M., Kratochvíl, J., Loebl, M., Matoušek, J., Valtr, P., Thomas, R. (eds) Topics in Discrete Mathematics. Algorithms and Combinatorics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-33700-8_15
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DOI: https://doi.org/10.1007/3-540-33700-8_15
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