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On the subgraph query problem

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  27 July 2020

Ryan Alweiss ,
Chady Ben Hamida ,
Xiaoyu He  and
Alexander Moreira
Ryan Alweiss*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08541, USA
Chady Ben Hamida
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08541, USA
Xiaoyu He
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA
Alexander Moreira
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08541, USA
*
*Corresponding author. Email: alweiss@math.princeton.edu
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Abstract

Given a fixed graph H, a real number p (0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d), where d is the degeneracy of H, by exhibiting an algorithm that finds a copy of H in time O(p−d) as p goes to 0. Furthermore, we prove that there are 2-degenerate graphs which require p−2+o(1) queries, showing for the first time that there exist graphs H for which f(H, p) does not grow like a constant power of p−1 as p goes to 0. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz and Tetali by showing that for any δ < 2, there exists α < 2 such that one cannot find a clique of order α log2n in G(n, 1/2) in nδ queries.

MSC classification

Primary: 05C57: Games on graphs
Secondary: 05C80: Random graphs 05D10: Ramsey theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 1 , January 2021 , pp. 1 - 16
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Research supported by an NSF Graduate Research Fellowship.

‡

Research supported by an NSF Graduate Research Fellowship.

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