On the interplay between Babai and Cerny's conjectures
François Gonze, Vladimir V. Gusev, Balázs Szabolcs Gerencsér, Raphaël M. Jungers, Mikhail Vladimirovich Volkov · Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) · 2017
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 $\tfrac{n^2}{4} + o(n^2)$.