Implicit representation of sparse hereditary families

TL;DR

This paper investigates the size of implicit graph representations for hereditary families, showing that certain speed bounds guarantee polynomial-sized labels, and establishing tightness of these bounds.

Contribution

It proves that hereditary families with moderate speed bounds admit polynomial-size implicit representations, extending previous results and identifying tight bounds.

Findings

Hereditary families with speed up to 2^{(1/4 - ε)n^2} have implicit representations of size O(n^{1-1/d} log n).

Existence of hereditary families where this bound is tight up to a logarithmic factor.

Demonstrates a spectrum of implicit representation sizes based on the speed of hereditary families.

Abstract

For a hereditary family of graphs $\FF$, let $\FF_n$ denote the set of all members of $\FF$ on $n$ vertices. The speed of $\FF$ is the function $f(n)=|\FF_n|$. An implicit representation of size $\ell(n)$ for $\FF_n$ is a function assigning a label of $\ell(n)$ bits to each vertex of any given graph $G \in \FF_n$, 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 $\FF_n$ for any hereditary family $\FF$ with speed $2^{\Omega(n^2)}$ is $(1+o(1)) \log_2 |\FF_n|/n~(=\Theta(n))$. 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 $\eps>0$ there is an integer $d \geq 1$ so that if $\FF$ is a hereditary family with speed $f(n) \leq 2^{(1/4-\eps)n^2}$ then $\FF_n$ 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.