Optimal Adjacency Labels for Subgraphs of Cartesian Products

TL;DR

This paper develops optimal adjacency labeling schemes for subgraphs of Cartesian products of graphs within a hereditary class, leveraging advanced techniques to extend efficiency results and improve upon recent research.

Contribution

It introduces a method to construct optimal adjacency labels for subgraphs of Cartesian products based on hereditary graph classes, extending existing efficiency results.

Findings

Achieves optimal adjacency labels for subgraphs of Cartesian products.

Extends efficiency results from base classes to their Cartesian products.

Uses advanced techniques from communication complexity, hashing, and combinatorics.

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].