Counting symbol switches in synchronizing automata

Abstract: Instead of looking at the lengths of synchronizing words as in \v{C}ern\'y's conjecture, we look at the switch count of such words, that is, we only count the switches from one letter to another. Where the synchronizing words of the \v{C}ern\'y automata $\mathcal{C}_n$ have switch count linear in $n$, we wonder whether synchronizing automata exist for which every synchronizing word has quadratic switch count. The answer is positive: we prove that switch count has the same complexity as synchronizing word length. We give some series of synchronizing automata yielding quadratic switch count, the best one reaching $\frac{2}{3} n² + O(n)$ as switch count. We investigate all binary automata on at most 9 states and determine the maximal possible switch count. For all $3\leq n\leq 9$, a strictly higher switch count can be reached by allowing more symbols. This behaviour differs from length, where for every $n$, no automata are known with higher synchronization length than $\mathcal{C}_n$, which has only two symbols. It is not clear if this pattern extends to larger $n$. For $n\geq 12$, our best construction only has two symbols.