Lower Bounds for the Length of Reset Words in Eulerian Automata

A complete deterministic finite automaton A is called synchronizing if the action of some word w resets A , that is, leaves the automaton in one particular state no matter at which state it is applied. Any such word w is said to be a reset word for the automaton. The minimum length of reset words for A is called the reset threshold of A and denoted by rt(A ). Synchronizing automata constitute an interesting combinatorial object and naturally appear in many applications such as coding theory, robotics and testing of reactive systems. For a brief introduction to the theory of synchronizing automata we refer the reader to the recent surveys [11, 16]. The interest to the field is also heated by the famous Černý conjecture. In 1964 Jan Černý [3] constructed for each n > 1 a synchronizing automaton Cn with n states whose reset threshold is (n− 1) . Soon after that he conjectured that these automata represent the worst possible case, that is, every synchronizing automaton with n states can be reset by a word of length (n − 1). Despite intensive research, the best upper bound on the reset threshold of synchronizing automata with n states achieved so far is n(7n +6n−16) 48 , see [15], so it is much larger