Block colourings of star systems

An $e$-star system of order $n$ is a decomposition of the complete graph $K_n$ into copies of the complete bipartite graph $K_{1,e}$ (or $e$-star). Such systems are known to exist if and only if $n\geq 2e$ and $e$ divides $\binom{n}{2}$. We consider block colourings of such systems, where each $e$-star is assigned a colour, and two $e$-stars which share a vertex receive different colours. We present a computer analysis of block colourings of small $3$-star systems. Furthermore, we prove that: (i) for $n\equiv 0,1$ mod $2e$ there exists either an $n$ or $(n-1)$-block colourable $e$-star system of order $n$; and (ii) when $e=3$, the same result holds in the remaining congruence classes mod $6$.