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

TL;DR

This paper disproves a long-standing conjecture in graph theory by constructing a counter-example using advanced communication complexity techniques, showing that certain hereditary graph properties do not admit small probabilistic universal graphs.

Contribution

The authors provide the first counter-example to the probabilistic universal graph conjecture, linking it to randomized communication complexity and matrix properties.

Findings

Counter-example refutes the conjecture.

Existence of matrices with unbounded complexity but bounded submatrix complexity.

Connects graph properties with communication complexity theory.

Abstract

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 it is stable and has at most factorial speed. Our counter-example follows from the existence of a sequence of $n \times n$ Boolean matrices $M_n$, such that their public-coin randomized communication complexity tends to infinity, while the randomized communication complexity of every $\sqrt{n}\times \sqrt{n}$ submatrix of $M_n$ is bounded by a universal constant.