A class of graphs admits an adjacency labeling scheme of size $f(n)$, if the vertices of any $n$-vertex graph $G$ in the class can be assigned binary strings (aka labels) of length $f(n)$ so that the adjacency between each pair of vertices in $G$ can be determined only from their labels. The Implicit Graph Conjecture (IGC) claimed that any graph class which is hereditary (i.e. closed under taking induced subgraphs) and factorial (i.e. containing $2^{\Theta(n \log n)}$ graphs on $n$ vertices) admits an adjacency labeling scheme of order optimal size $O(\log n)$. After thirty years open, the IGC was recently disproved [Hatami and Hatami, FOCS 2022]. In this work we show that the IGC does not hold even for monotone graph classes, i.e. classes closed under taking subgraphs. More specifically, we show that there are monotone factorial graph classes for which the size of any adjacency labeling scheme is $\Omega(\log^2 n)$. Moreover, this is best possible, as any monotone factorial class admits an adjacency labeling scheme of size $O(\log^2 n)$. This is a consequence of our general result that establishes tight bounds on the size of adjacency labeling schemes for monotone graph classes: for any function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ with $\log x \leq f(x) \leq x^{1-\delta}$ for some constant $\delta > 0$, that satisfies some natural conditions, there exist monotone graph classes, in which the number of $n$-vertex graphs grows as $2^{O(nf(n))}$ and that do not admit adjacency labels of size at most $f(n) \log n$. On the other hand any such class admits adjacency labels of size $O(f(n)\log n)$, which is a factor of $\log n$ away from the order optimal bound $O(f(n))$. This is the first example of tight bounds on adjacency labels for graph classes that do not admit order optimal adjacency labeling schemes.

翻译：一类图若满足：对于该类中任意$n$个顶点的图$G$，可为其顶点分配长度为$f(n)$的二进制串（即标签），使得$G$中任意两顶点间的邻接关系仅由它们的标签即可确定，则称该类图具有大小为$f(n)$的邻接标签方案。隐式图猜想（IGC）曾断言：任何遗传性（即对诱导子图封闭）且阶乘性（即$n$个顶点上的图数为$2^{\Theta(n \log n)}$）的图类均存在大小为最优阶$O(\log n)$的邻接标签方案。在悬而未决三十年后，该猜想近期被推翻[Hatami与Hatami，FOCS 2022]。本文证明IGC对单调图类（即对子图封闭的类）亦不成立。具体而言，我们构建了单调阶乘图类，其任意邻接标签方案的大小均为$\Omega(\log^2 n)$。更关键的是，这一下界是紧的——任何单调阶乘类均存在大小为$O(\log^2 n)$的邻接标签方案。该结论源于我们建立的单调图类邻接标签大小紧界的一般性结果：对任意满足$\log x \leq f(x) \leq x^{1-\delta}$（$\delta > 0$为常数）且满足若干自然条件的函数$f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$，存在单调图类，其中$n$顶点图的数量以$2^{O(nf(n))}$增长，且该类不存在大小不超过$f(n) \log n$的邻接标签；另一方面，任何此类图类均存在大小为$O(f(n)\log n)$的邻接标签，该结果与最优阶界$O(f(n))$相差一个$\log n$因子。这是首个关于不具备最优阶邻接标签方案的图类建立的邻接标签紧界实例。