  A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a
set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected
by a properly colored directed path in $D$, and (ii) every vertex outside $S$
can reach $S$ by a properly colored directed path in $D$. In this paper, we
conjecture that every arc-colored digraph with all cycles properly colored has
such a kernel and verify the conjecture for unicyclic digraphs, semi-complete
digraphs and bipartite tournaments, respectively. Moreover, weaker conditions
for the latter two classes of digraphs are given.
