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Large complete minors in random subgraphs

Part of: Graph theory

Published online by Cambridge University Press:  03 December 2020

Joshua Erde ,
Mihyun Kang  and
Michael Krivelevich
Joshua Erde*
Affiliation:
Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Mihyun Kang
Affiliation:
Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Michael Krivelevich
Affiliation:
School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv6997801, Israel
*
*Corresponding author. Email: erde@tugraz.at
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Abstract

Let G be a graph of minimum degree at least k and let Gp be the random subgraph of G obtained by keeping each edge independently with probability p. We are interested in the size of the largest complete minor that Gp contains when p = (1 + ε)/k with ε > 0. We show that with high probability Gp contains a complete minor of order $\tilde{\Omega}(\sqrt{k})$ , where the ~ hides a polylogarithmic factor. Furthermore, in the case where the order of G is also bounded above by a constant multiple of k, we show that this polylogarithmic term can be removed, giving a tight bound.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C83: Graph minors
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 4 , July 2021 , pp. 619 - 630
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Supported by the Austrian Science Fund (FWF): I3747.

‡

Supported in part by USA–Israel BSF grant 2018267, and by ISF grant 1261/17.

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