On state complexity of unions of binary factor-free languages

Abstract: It has been conjectured in 2011 by Brzozowski et al. that if $K$ and $L$ are factor-free regular languages over a binary alphabet having state complexity $m$ and $n$, resp, then the state complexity of $K\cup L$ is at most $mn-(m+n)+3-\min{m,n}$. We disprove this conjecture by giving a lower bound of $mn-(m+n)-2-\lfloor\frac{\min{m,n}-2}{2}\rfloor$, which exceeds the conjectured bound whenever $\min{m,n}\geq 10$.