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Edge-statistics on large graphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  14 November 2019

Noga Alon ,
Dan Hefetz ,
Michael Krivelevich  and
Mykhaylo Tyomkyn
Noga Alon
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey, USA School of Mathematics and Computer Science, Tel Aviv University, Tel Aviv, Israel
Dan Hefetz
Affiliation:
Department of Computer Science, Ariel University, Ariel40700, Israel
Michael Krivelevich
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv6997801, Israel
Mykhaylo Tyomkyn*
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv6997801, Israel Department of Mathematics, California Institute of Technology, Pasadena, CA91125
*
*Corresponding author. Email: tyomkyn@caltech.edu
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Abstract

The inducibility of a graph H measures the maximum number of induced copies of H a large graph G can have. Generalizing this notion, we study how many induced subgraphs of fixed order k and size ℓ a large graph G on n vertices can have. Clearly, this number is $\left( {\matrix{n \cr k}}\right)$ for every n, k and $\ell \in \left\{ {0,\left( {\matrix{k \cr 2}} \right)}\right\}$. We conjecture that for every n, k and $0 \lt \ell \lt \left( {\matrix{k \cr 2}}\right)$ this number is at most $ (1/e + {o_k}(1)) {\left( {\matrix{n \cr k}} \right)}$. If true, this would be tight for ℓ ∈ {1, k − 1}.

In support of our ‘Edge-statistics Conjecture’, we prove that the corresponding density is bounded away from 1 by an absolute constant. Furthermore, for various ranges of the values of ℓ we establish stronger bounds. In particular, we prove that for ‘almost all’ pairs (k, ℓ) only a polynomially small fraction of the k-subsets of V(G) have exactly ℓ edges, and prove an upper bound of $ (1/2 + {o_k}(1)){\left( {\matrix{n \cr k}}\right)}$ for ℓ = 1.

Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun’s sieve, as well as graph-theoretic and combinatorial arguments such as Zykov’s symmetrization, Sperner’s theorem and various counting techniques.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 2 , March 2020 , pp. 163 - 189
Copyright
© Cambridge University Press 2019

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Footnotes

†

Research supported in part by NSF grant DMS-1855464, ISF grant 281/17, BSF grant 2018267 and the Simons Foundation.

‡

The research of Dan Hefetz is supported by ISF grant 822/18.

§

Partially supported by USA-Israel BSF grants 2014361 and 2018267, and by ISF grant 1261/17.

The research of Mykhaylo Tyomkyn is supported in part by ERC Starting Grant 633509.

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