Optimal Adjacency Labels for Subgraphs of Cartesian Products

Authors Louis Esperet , Nathaniel Harms , Viktor Zamaraev

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Author Details

Louis Esperet
  • Laboratoire G-SCOP, Grenoble, France
Nathaniel Harms
  • EPFL, Lausanne, Switzerland
Viktor Zamaraev
  • University of Liverpool, UK


We are very grateful to Sebastian Wild, who prevented us trying to reinvent perfect hashing.

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Louis Esperet, Nathaniel Harms, and Viktor Zamaraev. Optimal Adjacency Labels for Subgraphs of Cartesian Products. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 57:1-57:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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 and hashing, and improves upon recent results of Chepoi, Labourel, and Ratel [Journal of Graph Theory, 2020].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Combinatorics
  • Adjacency labeling schemes
  • Cartesian product
  • Hypercubes


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