  We study dynamic conflict-free colorings in the plane, where the goal is to
maintain a conflict-free coloring (CF-coloring for short) under insertions and
deletions.
  - First we consider CF-colorings of a set $S$ of unit squares with respect to
points. Our method maintains a CF-coloring that uses $O(\log n)$ colors at any
time, where $n$ is the current number of squares in $S$, at the cost of only
$O(\log n)$ recolorings per insertion or deletion of a square. We generalize
the method to rectangles whose sides have lengths in the range $[1,c]$, where
$c$ is a fixed constant. Here the number of used colors becomes $O(\log^2 n)$.
The method also extends to arbitrary rectangles whose coordinates come from a
fixed universe of size $N$, yielding $O(\log^2 N \log^2 n)$ colors. The number
of recolorings for both methods stays in $O(\log n)$.
  - We then present a general framework to maintain a CF-coloring under
insertions for sets of objects that admit a unimax coloring with a small number
of colors in the static case. As an application we show how to maintain a
CF-coloring with $O(\log^3 n)$ colors for disks (or other objects with linear
union complexity) with respect to points at the cost of $O(\log n)$ recolorings
per insertion. We extend the framework to the fully-dynamic case when the
static unimax coloring admits weak deletions. As an application we show how to
maintain a CF-coloring with $O(\sqrt{n} \log^2 n)$ colors for points with
respect to rectangles, at the cost of $O(\log n)$ recolorings per insertion and
$O(1)$ recolorings per deletion. These are the first results on fully-dynamic
CF-colorings in the plane, and the first results for semi-dynamic CF-colorings
for non-congruent objects.
