The Cerny conjecture and (n-1)-Hamiltonian automata

A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. Černý conjectured in 1964 that a synchronizing automaton with n states has a synchronizing word of length at most (n− 1). We introduce the notion of aperiodically 1−contracting automata and prove that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Furthermore, we give a sufficient condition under which the Černý conjecture holds for aperiodically 1−contracting automata. As a special case, we prove some results for circular automata.