Tight bounds on adjacency labels for monotone graph classes

A class of graphs admits an adjacency labeling scheme of size $b(n)$, if the vertices in each of its $n$-vertex graphs can be assigned binary strings (called labels) of length $b(n)$ so that the adjacency of two vertices can be determined solely from their labels. We give tight bounds on the size of adjacency labels for every family of monotone (i.e., subgraph-closed) classes with a well-behaved growth function between $2^{O(n \log n)}$ and $2^{O(n^{2-\delta})}$ for any $\delta > 0$. Specifically, we show that for any function $f: \mathbb N \to \mathbb R$ satisfying $\log n \leqslant f(n) \leqslant n^{1-\delta}$ for any fixed $\delta > 0$, and some~sub-multiplicativity condition, there are monotone graph classes with growth $2^{O(nf(n))}$ that do not admit adjacency labels of size at most $f(n) \log n$. On the other hand, any such class does admit adjacency labels of size $O(f(n)\log n)$. Surprisingly this tight bound is a $\Theta(\log n)$ factor away from the information-theoretic bound of $\Omega(f(n))$. The special case when $f = \log$ implies that the recently-refuted Implicit Graph Conjecture [Hatami and Hatami, FOCS 2022] also fails within monotone classes. We further show that the Implicit Graph Conjecture holds for all monotone \emph{small} classes. In other words, any monotone class with growth rate at most $n!\,c^n$ for some constant $c>0$, admits adjacency labels of information-theoretic order optimal size. In fact, we show a more general result that is of independent interest: any monotone small class of graphs has bounded degeneracy.We conjecture that the Implicit Graph Conjecture holds for all hereditary small classes.