An adjacency sketching or implicit labeling scheme for a family $\cal F$ of graphs is a method that defines for any $n$ vertex $G \in \cal F$ an assignment of labels to each vertex in $G$, so that the labels of two vertices tell you whether or not they are adjacent. The goal is to come up with labeling schemes that use as few bits as possible to represent the labels. By using randomness when assigning labels, it is sometimes possible to produce adjacency sketches with much smaller label sizes, but this comes at the cost of introducing some probability of error. Both deterministic and randomized labeling schemes have been extensively studied, as they have applications for distributed data structures and deeper connections to universal graphs and communication complexity. The main question of interest is which graph families have schemes using short labels, usually $O(\log n)$ in the deterministic case or constant for randomized sketches. In this work we consider the resilience of probabilistic adjacency sketches against an adversary making adaptive queries to the labels. This differs from the previously analyzed probabilistic setting which is ``one shot". We show that in the adaptive adversarial case the size of the labels is tightly related to the maximal degree of the graphs in $\cal F$. This results in a stronger characterization compared to what is known in the non-adversarial setting. In more detail, we construct sketches that fail with probability $\varepsilon$ for graphs with maximal degree $d$ using $2d\log (1/\varepsilon)$ bit labels and show that this is roughly the best that can be done for any specific graph of maximal degree $d$, e.g.\ a $d$-ary tree.

翻译：对于图族$\cal F$，邻接草图或隐式标记方案是一种方法，该方法为任意$n$个顶点的图$G \in \cal F$中每个顶点分配标签，使得两个顶点的标签能判断它们是否相邻。目标在于设计使用尽可能少比特表示标签的标记方案。通过随机分配标签，有时可以生成标签尺寸更小的邻接草图，但这会以引入一定错误概率为代价。确定性和随机性标记方案已被广泛研究，因其在分布式数据结构中的应用，以及与通用图和通信复杂度的深层联系。核心问题在于哪些图族能够使用短标签方案（确定性情形下通常为$O(\log n)$，随机草图中为常数）。本研究考虑概率性邻接草图在对抗性自适应查询标签环境中的鲁棒性。这与先前分析的“一次性”概率设置不同。我们证明在自适应对抗情形下，标签尺寸与图族$\cal F$中图的最大度紧密相关，从而形成比非对抗环境中更强的刻画特征。具体而言，我们针对最大度为$d$的图构造了错误概率为$\varepsilon$的草图，使用$2d\log (1/\varepsilon)$比特标签，并证明对于任意最大度为$d$的特定图（例如$d$叉树），这几乎是最优结果。