  In this paper we study the problem of coloring a unit interval graph which
changes dynamically. In our model the unit intervals are added or removed one
at the time, and have to be colored immediately, so that no two overlapping
intervals share the same color. After each update only a limited number of
intervals is allowed to be recolored. The limit on the number of recolorings
per update is called the recourse budget. In this paper we show, that if the
graph remains $k$-colorable at all times, and the updates consist of insertions
only, then we can achieve the amortized recourse budget of $O(k^7 \log n)$
while maintaining a proper coloring with $k$ colors. This is an exponential
improvement over the result in [Bosek et al., Recoloring Interval Graphs with
Limited Recourse Budget. SWAT 2020] in terms of both $k$ and $n$. We complement
this result by showing the lower bound of $\Omega(n)$ on the amortized recourse
budget in the fully dynamic setting. Our incremental algorithm can be
efficiently implemented.
  As a byproduct of independent interest we include a new result on coloring
proper circular arc graphs. Let $L$ be the maximum number of arcs intersecting
in one point for some set of unit circular arcs $\mathcal{A}$. We show that if
there is a set $\mathcal{A}'$ of non-intersecting unit arcs of size $L^2-1$
such that $\mathcal{A} \cup \mathcal{A}'$ does not contain $L+1$ arcs
intersecting in one point, then it is possible to color $\mathcal{A}$ with $L$
colors. This complements the work on unit circular arc coloring, which
specifies sufficient conditions needed to color $\mathcal{A}$ with $L+1$ colors
or more.
