A graph class admits an implicit representation 1 if, for every positive integer n, its n-vertex graphs have a b(n)-bit (adjacency) labeling scheme with b(n) = O(log n), i.e., their vertices can be labeled by binary strings of length b(n) such that the presence of an edge between any pair of vertices u, v can be deduced solely from the labels of u and v. 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 finally 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!c n 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 proven in the paper: Every weakly sparse small class has bounded expansion (hence, in particular, bounded degeneracy). The latter generalizes and strengthens the previous results that every monotone small class has bounded degeneracy [Bonnet et al., ICALP '24], and that every weakly sparse class of bounded twin-width has bounded expansion [Bonnet et al., Combinatorial Theory '22]. Second, we show that every hereditary small class admits an O(log 3 n)-bit labeling scheme, which provides a substantial improvement over the best-known polynomial upper bound of n 1-ε on the size of adjacency labeling schemes for such classes. To do so, we establish that every small class has neighborhood complexity O(n log n), also of independent interest. We then apply a classic result, due to Welzl [SoCG '88], on efficiently ordering the universe of a set system of low Vapnik-Chervonenkis dimension such that every set can be described as the union of a limited number of intervals along this order.