Small But Unwieldy: A Lower Bound on Adjacency Labels for Small Classes

TL;DR

This paper proves that for any small class of graphs, there is a lower bound on adjacency label sizes, showing that tiny classes can lack compact universal graphs and challenging existing conjectures.

Contribution

It establishes the existence of small monotone graph classes without small adjacency labeling schemes, disproving the implicit graph conjecture and related hypotheses.

Findings

Existence of small classes without adjacency labels of size s log n

Small classes can have unbounded twin-width

Disproof of the Small conjecture with a self-contained proof

Abstract

We show that for any natural number $s$, there is a constant $\gamma$ and a subgraph-closed class having, for any natural $n$, at most $\gamma^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size at most $s \log n$. In other words, for every $s$, there is a small (even tiny) monotone class without universal graphs of size $n^s$. Prior to this result, it was not excluded that every small class has an almost linear universal graph, or equivalently a labeling scheme with labels of size $(1+o(1))\log n$. The existence of such a labeling scheme, a scaled-down version of the recently disproved Implicit Graph Conjecture, was repeatedly raised [Gavoille and Labourel, ESA '07; Dujmovi\'{c} et al., JACM '21; Bonamy et al., SIDMA '22; Bonnet et al., Comb. Theory '22]. Furthermore, our small monotone classes have unbounded twin-width, thus simultaneously disprove the already-refuted Small conjecture; but this time with a self-contained proof, not relying on elaborate group-theoretic constructions.