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-multiplicative 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 $O(f(n))$. The special case when $f = \log$ implies that the recently-refuted Implicit Graph Conjecture [Hatami and Hatami, FOCS 2022] also fails within monotone classes.

翻译：图类在$b(n)$规模下允许邻接标记方案，若其每个$n$顶点图中的顶点可被分配长度为$b(n)$的二进制字符串（称为标签），使得两顶点是否相邻仅由它们的标签决定。我们针对所有具有良性增长函数的单调（即子图封闭）图类族，给出了邻接标签规模的紧界，该增长函数介于$2^{O(n \log n)}$与$2^{O(n^{2-\delta})}$之间（对任意$\delta > 0$）。具体而言，我们证明：对于任意满足$\log n \leqslant f(n) \leqslant n^{1-\delta}$（固定$\delta > 0$）及某种次乘性条件的函数$f: \mathbb N \to \mathbb R$，存在增长率为$2^{O(nf(n))}$的单调图类，其不允许规模至多为$f(n)\log n$的邻接标签。另一方面，此类图类确实允许规模为$O(f(n)\log n)$的邻接标签。令人惊讶的是，该紧界与信息论下界$O(f(n))$相差一个$\Theta(\log n)$因子。当$f = \log$的特殊情况表明，近期被反驳的隐式图猜想[Hatami and Hatami, FOCS 2022]在单调类中同样不成立。