Distinguishing Tournaments with Small Label Classes

A $d$-distinguishing vertex (arc) labeling of a digraph is a vertex (arc) labeling using $d$ labels that is not preserved by any nontrivial automorphism. Let $\rho(T)$ ($\rho'(T)$) be the minimum size of a label class in a 2-distinguishing vertex (arc) labeling of a tournament $T$. Gluck's Theorem implies that $\rho(T) \le \lfloor n/2 \rfloor$ for any tournament $T$ of order $n$. In this paper we construct a family of tournaments $\cal H$ such that $\rho(T) \ge \lfloor n/2 \rfloor$ for any order $n$ tournament in $\cal H$. Additionally, we prove that $\rho'(T) \le \lfloor 7n/36 \rfloor + 3$ for any tournament $T$ of order $n$ and $\rho'(T) \ge \lceil n/6 \rceil$ when $T \in {\cal H}$ and has order $n$. These results answer some open questions stated by Boutin.