On the edit distance function of the random graph

TL;DR

This paper investigates the asymptotic behavior of the edit distance function for graphs forbidding a specific Erdős-Rényi random graph, revealing precise formulas within certain parameter ranges.

Contribution

It provides a new asymptotic formula for the edit distance function in the context of hereditary graph properties related to forbidden Erdős-Rényi subgraphs.

Findings

The edit distance function scales as (2 log n)/n times a minimum of two terms.

The formula applies asymptotically for large n_0 when p_0 is within a specific interval.

Results hold for p in [1/3, 2/3] for any p_0 in (0,1).

Abstract

Given a hereditary property of graphs $\mathcal{H}$ and a $p\in [0,1]$, the edit distance function ${\rm ed}_{\mathcal{H}}(p)$ is asymptotically the maximum proportion of edge-additions plus edge-deletions applied to a graph of edge density $p$ sufficient to ensure that the resulting graph satisfies $\mathcal{H}$. The edit distance function is directly related to other well-studied quantities such as the speed function for $\mathcal{H}$ and the $\mathcal{H}$-chromatic number of a random graph. Let $\mathcal{H}$ be the property of forbidding an Erd\H{o}s-R\'{e}nyi random graph $F\sim \mathbb{G}(n_0,p_0)$, and let $\varphi$ represent the golden ratio. In this paper, we show that if $p_0\in [1-1/\varphi,1/\varphi]$, then a.a.s. as $n_0\to\infty$, \begin{align*} {\rm ed}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{-\log(1-p_0)}, \frac{1-p}{-\log p_0} \right\}. \end{align*} Moreover, this holds for $p\in [1/3,2/3]$ for any $p_0\in (0,1)$.