Reset Thresholds of Automata with Two Cycle Lengths

Abstract

We present several series of synchronizing automata with multiple parameters, generalizing previously known results. Let p and q be two arbitrary co-prime positive integers, [Formula: see text]. We establish reset thresholds of the colorings of primitive digraphs with exactly one simple cycle of length p and one simple cycle of length q. Also, we study reset thresholds of the colorings of primitive digraphs with exactly one simple cycle of length q and two simple cycles of length p. We obtain the reset thresholds of the form [Formula: see text], where [Formula: see text] is a linear function of p and q.