Edgestatistics on large graphs
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 $\ell$ a large graph $G$ on $n$ vertices can have. Clearly, this number is $\binom{n}{k}$ for every $n$, $k$ and $\ell \in \left \{0, \binom{k}{2} \right\}$. We conjecture that for every $n$, $k$ and $0 < \ell < \binom{k}{2}$ this number is at most $\left(1/e + o_k(1) \right) \binom{n}{k}$. If true, this would be tight for $\ell \in \{1, k1\}$. In support of our `Edgestatistics conjecture' we prove that the corresponding density is bounded away from $1$ by an absolute constant. Furthermore, for various ranges of the values of $\ell$ we establish stronger bounds. In particular, we prove that for `almost all' pairs $(k, \ell)$ only a polynomially small fraction of the $k$subsets of $V(G)$ has exactly $\ell$ edges, and prove an upper bound of $(1/2 + o_k(1))\binom{n}{k}$ for $\ell = 1$. Our proof methods involve probabilistic tools, such as anticoncentration results relying on fourth moment estimates and Brun's sieve, as well as graphtheoretic and combinatorial arguments such as Zykov's symmetrization, Sperner's theorem and various counting techniques.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 DOI:
 10.48550/arXiv.1805.06848
 arXiv:
 arXiv:1805.06848
 Bibcode:
 2018arXiv180506848A
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability
 EPrint:
 23 pages, revised version