# On the interplay between Babai and Cerny's conjectures

## TL;DR

This paper investigates the maximum reset thresholds and diameters of certain automata related to the Babai and Cerny conjectures, providing new bounds and constructions for automata with specific properties.

## Contribution

It establishes upper bounds on reset thresholds for automata with full transformation monoids and constructs permutation automata with large diameters, advancing understanding of these conjectures.

## Key findings

- Reset thresholds are bounded by 2n^2 - 6n + 5.
- Reset thresholds can reach n(n-1)/2.
- Constructed permutation automata with diameter approximately n^2/4.

## Abstract

Motivated by the Babai conjecture and the Cerny conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with $n$ states in this class, we prove that the reset thresholds are upper-bounded by $2n^2-6n+5$ and can attain the value $\tfrac{n(n-1)}{2}$. In addition, we study diameters of the pair digraphs of permutation automata and construct $n$-state permutation automata with diameter $\tfrac{n^2}{4} + o(n^2)$.