Implicit Representation of Sparse Hereditary Families

Abstract

For a hereditary family of graphs ${{\mathcal {F}}}$, let ${{\mathcal {F}}}_n$ denote the set of all members of ${{\mathcal {F}}}$ on n vertices. The speed of ${{\mathcal {F}}}$ is the function $f(n)=|{{\mathcal {F}}}_n|$. An implicit representation of size $\ell (n)$ for ${{\mathcal {F}}}_n$ is a function assigning a label of $\ell (n)$ bits to each vertex of any given graph $G\in {{\mathcal {F}}}_n$, so that the adjacency between any pair of vertices can be determined by their labels. Bonamy, Esperet, Groenland, and Scott proved that the minimum possible size of an implicit representation of ${{\mathcal {F}}}_n$ for any hereditary family ${{\mathcal {F}}}$ with speed $2^{\Omega (n^2)}$ is $(1+o(1))\log _2|{{\mathcal {F}}}_n|/n$ ($=\Theta (n)$). A recent result of Hatami and Hatami shows that the situation is very different for very sparse hereditary families. They showed that for every $\delta >0$ there are hereditary families of graphs with speed $2^{O(n\log n)}$ that do not admit implicit representations of size smaller than $n^{1/2-\delta }$. In this note we show that even a mild speed bound ensures an implicit representation of size $O(n^c)$ for some $c<1$. Specifically we prove that for every ${\varepsilon }>0$ there is an integer $d\ge 1$ so that if ${{\mathcal {F}}}$ is a hereditary family with speed $f(n) \le 2^{(1/4-{\varepsilon })n^2}$ then ${{\mathcal {F}}}_n$ admits an implicit representation of size $O(n^{1-1/d}\log n)$. Moreover, for every integer $d>1$ there is a hereditary family for which this is tight up to the logarithmic factor.