An Extremal Series of Eulerian Synchronizing Automata

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