TY - JOUR
T1 - Large complete minors in random subgraphs
AU - Erde, Joshua
AU - Kang, Mihyun
AU - Krivelevich, Michael
N1 - Publisher Copyright:
© The Author(s), 2020. Published by Cambridge University Press.
PY - 2021/7
Y1 - 2021/7
N2 - 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 Ω~ (√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.
AB - 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 Ω~ (√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.
UR - http://www.scopus.com/inward/record.url?scp=85097267514&partnerID=8YFLogxK
U2 - 10.1017/S0963548320000607
DO - 10.1017/S0963548320000607
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AN - SCOPUS:85097267514
SN - 0963-5483
VL - 30
SP - 619
EP - 630
JO - Combinatorics Probability and Computing
JF - Combinatorics Probability and Computing
IS - 4
ER -