  In an undirected graph, a conflict-free coloring (with respect to open
neighborhoods) is an assignment of colors to the vertices of the graph $G$ such
that every vertex in $G$ has a uniquely colored vertex in its open
neighborhood. The conflict-free coloring problem asks to find the smallest
number of colors required for a conflict-free coloring.
  The conflict-free coloring problem is NP-complete. From results in Abel et.
al. [SODA 2017], it can be inferred that every planar graph has a conflict-free
coloring with at most nine colors. As the best known lower bound for planar
graphs is four colors, it was asked in the same paper if fewer colors would
suffice. We make progress in answering this question, by showing that every
planar graph can be colored using at most six colors. The same proof idea is
used to show that every outerplanar graph can be colored using at most five
colors. Using a different approach, we further show that every outerplanar
graph can be colored using at most four colors.
  Finally, we study the problem on Kneser graphs. We show that $k+2$ colors are
necessary and sufficient to color the Kneser graph $K(n,k)$ when $n\geq
k(k+1)^2 + 1$.
