Abstract
Given a hereditary property of graphs
$\mathcal{H}$
and a
$p\in [0,1]$
, the edit distance function
$\textrm{ed}_{\mathcal{H}}(p)$
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 Erdős–Rényi 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*} {\textrm{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
$p_0\in (0,1)$
.
A primary tool in the proof is the categorization of p-core coloured regularity graphs in the range
$p\in[1-1/\varphi,1/\varphi]$
. Such coloured regularity graphs must have the property that the non-grey edges form vertex-disjoint cliques.
The edit distance function for graphs: an exploration of the case of forbidden induced K2,t and other questions
