Optimal Adjacency Labels for Subgraphs of Cartesian Products
Louis Esperet, Nathaniel Harms, Viktor Zamaraev
TL;DR
The paper develops optimal adjacency labeling schemes for subgraphs and induced subgraphs of Cartesian products of graphs from a hereditary class $\mathcal{F}$, connecting product-closure labeling to the base class via $s(n)$-size schemes and information-theoretic limits. It introduces a three-phase framework: (1) exactly one-coordinate differences, (2) XOR-labeling to compose per-factor labels, and (3) minimal perfect hashing to account for edge deletions, yielding additive $O(\log n)$ overheads and, in the induced-subgraph case, $O(\delta(n))$ overheads. The main results show $\mathsf{her}(\mathcal{F}^\square)$ admits labeling size $2s(n)+O(\log n)$ and $\mathsf{mon}(\mathcal{F}^\square)$ admits $s(n)+O(\delta(n)+\log n)$, tight up to constants, with corollaries that efficient labeling transfers from $\mathcal{F}$ to its Cartesian-product closures. These findings refine the understanding of the implicit-graph conjecture in the Cartesian-product setting and provide explicit, efficient encoders/decoders by combining communication complexity, hashing, and additive-combinatorics techniques.
Abstract
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].