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Recent Trends in Combinatorics pp 235–256Cite as

Giant components in random graphs

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

The phase transition in random graphs was first discussed by Erdős and Rényi who showed that a random graph undergoes a drastic change in the order and structure of the largest component. In view of recent results, one can recognise two main trends that have increased our understanding of random graphs. The first trend is the study of various random graph models (e.g. random hypergraphs, random graph processes, random graphs with degree constraints, random graphs on surfaces). The second trend is finding new simple proofs (and thereby improvement) of classical results on random graphs. This article discusses developments in random graphs in the light of these trends, with focus on giant components, limit theorems, and proof techniques.

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Acknowledgements

This article is based on the author’s lecture at the IMA Workshop “Probabilistic and Extremal Combinatorics”, which took place at the Institute of Mathematics and its Applications (IMA), University of Minnesota, 8–12 September 2014. The author is supported by the Austrian Science Fund (FWF): P26826, W1230.

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Authors and Affiliations

  1. Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010, Graz, Austria

    Mihyun Kang

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  1. Mihyun Kang
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Correspondence to Mihyun Kang .

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Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Kang, M. (2016). Giant components in random graphs. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_10

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