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Adjacency Labeling Schemes for Small Classes

Authors Édouard Bonnet , Julien Duron , John Sylvester , Viktor Zamaraev

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Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Julien Duron
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
John Sylvester
  • Department of Computer Science, University of Liverpool, UK
Viktor Zamaraev
  • Department of Computer Science, University of Liverpool, UK

Funding

This work has been supported by the Royal Society (IESbackslash R1backslash 231083), by the ANR projects TWIN-WIDTH (ANR-21-CE48-0014) and Digraphs (ANR-19-CE48-0013), and also the EPSRC project EP/T004878/1: Multilayer Algorithmics to Leverage Graph Structure.

Acknowledgements

We are grateful to Maksim Zhukovskii for many stimulating discussions on the topic of this paper. The first author thanks Szymon Toruńczyk for introducing him to (and answering questions on) the topic of paths with low crossing number.

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Édouard Bonnet, Julien Duron, John Sylvester, and Viktor Zamaraev. Adjacency Labeling Schemes for Small Classes. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 21:1-21:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.21

Abstract

A graph class admits an implicit representation if, for every positive integer n, its n-vertex graphs have a O(log n)-bit (adjacency) labeling scheme, i.e., their vertices can be labeled by binary strings of length O(log n) such that the presence of an edge between any pair of vertices can be deduced solely from their labels. The famous Implicit Graph Conjecture posited that every hereditary (i.e., closed under taking induced subgraphs) factorial (i.e., containing 2^O(n log n) n-vertex graphs) class admits an implicit representation. The conjecture was recently refuted [Hatami and Hatami, FOCS '22], and does not even hold among monotone (i.e., closed under taking subgraphs) factorial classes [Bonnet et al., ICALP '24]. However, monotone small (i.e., containing at most n! cⁿ many n-vertex graphs for some constant c) classes do admit implicit representations.
This motivates the Small Implicit Graph Conjecture: Every hereditary small class admits an O(log n)-bit labeling scheme. We provide evidence supporting the Small Implicit Graph Conjecture. First, we show that every small weakly sparse (i.e., excluding some fixed bipartite complete graph as a subgraph) class has an implicit representation. This is a consequence of the following fact of independent interest proved in the paper: Every weakly sparse small class has bounded expansion (hence, in particular, bounded degeneracy). Second, we show that every hereditary small class admits an O(log³ n)-bit labeling scheme, which provides a substantial improvement of the best-known polynomial upper bound of n^(1-ε) on the size of adjacency labeling schemes for such classes. This is a consequence of another fact of independent interest proved in the paper: Every small class has neighborhood complexity O(n log n).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • Adjacency labeling
  • degeneracy
  • weakly sparse classes
  • small classes
  • implicit graph conjecture

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