  A $P_m$ path in a graph is a path on $m$ vertices. A $P_m$ system of order
$n>1$ is a partition of the edges of the complete graph $K_n$ into $P_m$ paths.
A $P_m$ system is said to be $k$-colourable if the vertex set of $K_n$ can be
partitioned into $k$ sets called colour classes such that no path in the system
is monochromatic. The system is $k$-chromatic if it is $k$-colourable but is
not $(k-1)$-colourable. If every $k$-colouring of a $P_m$ system can be
obtained from some $k$-colouring $\phi$ by a permutation of the colours, we say
that the system is uniquely $k$-colourable. In this paper, we first observe
that there exists a $k$-chromatic $P_m$ system for any $k\geq 2$ and $m\geq 4$
where $m$ is even. Next, we prove that there exists an equitably 2-chromatic
$P_4$ system of order $n$ for each admissible order $n$. We then show that for
all $k\geq 3$, there exists a $k$-chromatic $P_4$ system of order $n$ for all
sufficiently large admissible $n$. Finally, we show that there exists a
uniquely 2-chromatic $P_4$ system of order $n$ for each admissible $n \geq
109$.
