Unique subgraphs are rare

TL;DR

This paper proves that as the number of vertices increases, the proportion of unique subgraphs in any graph tends to zero, confirming Erdős's intuition that such unique subgraphs are exceedingly rare.

Contribution

The paper establishes that the maximum proportion of unique subgraphs in any n-vertex graph approaches zero as n grows, resolving a long-standing question posed by Erdős.

Findings

f(n) tends to 0 as n increases

Unique subgraphs are asymptotically negligible in large graphs

Confirms Erdős's conjecture about rarity of unique subgraphs

Abstract

A folklore result attributed to P\'olya 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, Erd\H{o}s asked whether there exists $\delta>0$ such that $f(n)>\delta$ 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 Erd\H{o}s' 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.