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Upper bound for reset threshold of Eulerian synchronizing automata

Prove that for every n≥3, the reset threshold of any Eulerian synchronizing deterministic finite automaton with n states is at most ⌊(n^2−3)/2⌋.

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Background

Eulerian automata are those whose underlying directed graph is Eulerian. Kari established a general upper bound n2−3n+3 for their reset thresholds, and Szykuła–Vorel constructed extremal series achieving large lower bounds.

The authors report a conjecture by Szykuła and Vorel proposing the sharper universal upper bound ⌊(n2−3)/2⌋ for Eulerian synchronizing automata, with experimental verification for many small cases, yet the conjecture remains unproved in general.

References

Marek Szyku\l{}a and Vojt\v{e}ch Vorel have conjectured that for n\ge 3, \lfloor\frac{n2-3}2\rfloor is also an upper bound for the reset threshold of n-state Eulerian synchronizing automataConjecture 19. They report that exhaustive searches over small synchronizing Eulerian DFAs verified the bound for the following cases: DFAs with two letters and \le 11 states, with four letters and \le 7 states, with eight letters and \le 5 states, and all DFAs with at most 4 states.

— List of Results on the Černý Conjecture and Reset Thresholds for Synchronizing Automata (2508.15655 - Volkov, 21 Aug 2025) in Section A6, “Eulerian automata,” Comments