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.

翻译：一类图若允许大小为$b(n)$的邻接标记方案，则其每个$n$顶点图中的顶点可被赋予长度为$b(n)$的二进制串（称为标签），使得两顶点的邻接关系可仅凭其标签判定。针对所有单调（即子图封闭）图类，当增长函数在$2^{O(n\log n)}$至$2^{O(n^{2-\delta})}$（对任意$\delta>0$）之间表现良好时，我们给出其邻接标签大小的紧界。具体而言，对于满足$\log n \leqslant f(n) \leqslant n^{1-\delta}$（对任意固定$\delta>0$）及次可乘性条件的任意函数$f:\mathbb N \to \mathbb R$，存在增长率为$2^{O(nf(n))}$的单调图类，其无法接受大小至多为$f(n)\log n$的邻接标签。另一方面，此类图类均能接受大小为$O(f(n)\log n)$的邻接标签。令人惊讶的是，这一紧界与信息论下界$\Omega(f(n))$相差$\Theta(\log n)$因子。当$f=\log$这一特例表明，近期被证伪的隐式图猜想[Hatami and Hatami, FOCS 2022]在单调类中同样不成立。我们进一步证明隐式图猜想对所有单调*小*类成立：即对于增长率不超过$n!\,c^n$（$c$为某正常数）的任意单调图类，存在信息论阶数最优大小的邻接标签。实际上，我们证明了更具独立价值的更一般结论：任何单调小图类都具有有界退化性。我们猜想隐式图猜想对所有遗传小类成立。