# The Implicit Graph Conjecture is False

@article{Hatami2021TheIG, title={The Implicit Graph Conjecture is False}, author={Hamed Hatami and Pooya Hatami}, journal={ArXiv}, year={2021}, volume={abs/2111.13198} }

An efficient implicit representation of an n-vertex graph G in a family F of graphs assigns to each vertex of G a binary code of length O(log n) so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family but not on the individual graph. Every family of graphs admitting such a representation contains at most 2 log(n)) graphs on n vertices, and thus has at most factorial speed of growth. The Implicit Graph…

## 3 Citations

Implicit representation of sparse hereditary families

- MathematicsArXiv
- 2022

It is proved that for every ε > 0 there is an integer d ≥ 1 so that if F is a hereditary family with speed f(n) ≤ 2(1/4−ε)n2 then Fn admits an implicit representation of size O(n logn).

A counter-example to the probabilistic universal graph conjecture via randomized communication complexity

- MathematicsArXiv
- 2021

We refute the Probabilistic Universal Graph Conjecture of Harms, Wild, and Zamaraev, which states that a hereditary graph property admits a constant-size probabilistic universal graph if and only if…

Sketching Distances in Monotone Graph Classes

- Computer Science
- 2022

It is shown that, for monotone classes of graphs, there is a strict hierarchy: approximate distance threshold sketches imply small-distance sketches, which imply adjacency sketches, whereas the reverse implications are each false.

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