Unique subgraphs are rare

Abstract

A folklore result attributed to Pólya states that there are $(1 + o(1))2^{\binom{n}{2}}/n!$ non-isomorphic graphs on $n$ vertices. Given two graphs $G$ and $H$, we say that $G$ is a unique subgraph of $H$ if $H$ contains exactly one subgraph isomorphic to $G$. For an $n$-vertex graph $H$, let $f(H)$ be the number of non-isomorphic unique subgraphs of $H$ divided by $2^{\binom{n}{2}}/n!$ and let $f(n)$ denote the maximum of $f(H)$ over all graphs $H$ on $n$ vertices. In 1975, Erdős asked whether there exists $δ>0$ such that $f(n)>δ$ for all $n$ and offered $\$100$ for a proof and $\$25$ for a disproof, indicating he does not believe this to be true. We verify Erdős' intuition by showing that $f(n)\rightarrow 0$ as $n$ tends to infinity, i.e. no graph on $n$ vertices contains a constant proportion of all graphs on $n$ vertices as unique subgraphs.

Theorems & Definitions (12)

• Theorem 1.1: Pólya; Wright
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3: e.g. janson-luczak-ruczinski
• Lemma 2.4
• proof
• Claim 2.5