2023/07/20 by Édouard Bonnet, Bonnet, Édouard, Julien Duron +7
Computer Science · Engineering · Mathematics · #05C80 #68R01 #68R05 #Advanced Graph Theory Research #Combinatorics (math.CO) #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #G.2.1 #Limits and Structures in Graph Theory #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2307.11225
openalex publication_date 2023/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that for any natural number s, there is a constant γ and a subgraph-closed class having, for any natural n, at most γn 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 ns. 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; Dujmović 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.