  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].
