  In 2018, Bai, Fujita and Zhang (\emph{Discrete Math.} 2018, 341(6):
1523-1533) introduced the concept of a kernel by rainbow paths (for short,
RP-kernel) of an arc-coloured digraph $D$, which is a subset $S$ of vertices of
$D$ such that ($a$) there exists no rainbow path for any pair of distinct
vertices of $S$, and ($b$) every vertex outside $S$ can reach $S$ by a rainbow
path in $D$. They showed that it is NP-hard to recognize wether an arc-coloured
digraph has a RP-kernel and it is NP-complete to decided wether an arc-coloured
tournament has a RP-kernel. In this paper, we give the sufficient conditions
for the existence of a RP-kernel in arc-coloured unicyclic digraphs,
semicomplete digraphs, quasi-transitive digraphs and bipartite tournaments, and
prove that these arc-coloured digraphs have RP-kernels if certain "short"
cycles and certain "small" induced subdigraphs are rainbow.
