An Extremal Series of Eulerian Synchronizing Automata

TL;DR

This paper introduces an infinite series of Eulerian automata with long reset words, improving known lower bounds and suggesting a potential upper bound, supported by experimental verification.

Contribution

It presents a new infinite series of Eulerian automata with long reset words and conjectures this length as an upper bound, supported by exhaustive experiments.

Findings

Lower bound on reset word length is improved to (n^2-3)/2.

Experimental verification supports the conjecture that this bound is tight.

Provides a new perspective on the complexity of Eulerian automata.

Abstract

We present an infinite series of $n$-state Eulerian automata whose reset words have length at least $(n^2-3)/2$. This improves the current lower bound on the length of shortest reset words in Eulerian automata. We conjecture that $(n^2-3)/2$ also forms an upper bound for this class and we experimentally verify it for small automata by an exhaustive computation.