For any hereditary graph class $F$, we construct optimal adjacency labeling schemes for the classes of subgraphs and induced subgraphs of Cartesian products of graphs in $F$. As a consequence, we show that, if $F$ admits efficient adjacency labels (or, equivalently, small induced-universal graphs) meeting the information-theoretic minimum, then the classes of subgraphs and induced subgraphs of Cartesian products of graphs in $F$ do too. Our proof uses ideas from randomized communication complexity, hashing, and additive combinatorics, and improves upon recent results of Chepoi, Labourel, and Ratel [Journal of Graph Theory, 2020].

翻译：对于任何遗传图类 $F$，我们为 $F$ 中图的笛卡尔积的子图和诱导子图构造了最优邻接标签方案。作为推论，我们证明：如果 $F$ 具有满足信息论下界的高效邻接标签（或等价地，小型通用诱导图），那么 $F$ 中图的笛卡尔积的子图和诱导子图也具有该性质。我们的证明利用了随机通信复杂度、哈希和加性组合学中的思想，改进了 Chepoi、Labourel 和 Ratel 近期的工作 [Journal of Graph Theory, 2020]。