A Note on the Rainbow Connectivity of Tournaments

Abstract

An arc-coloured digraph $D$ is said to be \emph{rainbow connected} if for every two vertices $u$ and $v$ there is an $uv$-path all whose arcs have different colours. The minimun number of colours required to make the digraph rainbow connected is called the \emph{rainbow connection number} of $D$, denoted $\overrightarrow{\mathrm{rc}}(D)$. In \cite{Dorbec} it was showed that if $T$ is a strong tournament with $n\ge 5$ vertices, then $2\le \overrightarrow{\mathrm{rc}}(T)\le n-1$; and that for every $n$ and $k$ such that $3\le k\le n-1$, there exists a tournament $T$ on $n$ vertices such that $\overrightarrow{\mathrm{rc}}(T)=k$. In this note it is showed that for any $n\ge 6$, there is a tournament $T$ of $n$ vertices such that $\overrightarrow{\mathrm{rc}}(T)=2$.