  The distributed coloring problem is at the core of the area of distributed
graph algorithms and it is a problem that has seen tremendous progress over the
last few years. Much of the remarkable recent progress on deterministic
distributed coloring algorithms is based on two main tools: a) defective
colorings in which every node of a given color can have a limited number of
neighbors of the same color and b) list coloring, a natural generalization of
the standard coloring problem that naturally appears when colorings are
computed in different stages and one has to extend a previously computed
partial coloring to a full coloring.
  In this paper, we introduce 'list defective colorings', which can be seen as
a generalization of these two coloring variants. Essentially, in a list
defective coloring instance, each node $v$ is given a list of colors
$x_{v,1},\dots,x_{v,p}$ together with a list of defects $d_{v,1},\dots,d_{v,p}$
such that if $v$ is colored with color $x_{v, i}$, it is allowed to have at
most $d_{v, i}$ neighbors with color $x_{v, i}$.
  We highlight the important role of list defective colorings by showing that
faster list defective coloring algorithms would directly lead to faster
deterministic $(\Delta+1)$-coloring algorithms in the LOCAL model. Further, we
extend a recent distributed list coloring algorithm by Maus and Tonoyan [DISC
'20]. Slightly simplified, we show that if for each node $v$ it holds that
$\sum_{i=1}^p \big(d_{v,i}+1)^2 > \mathrm{deg}_G^2(v)\cdot polylog\Delta$ then
this list defective coloring instance can be solved in a
communication-efficient way in only $O(\log\Delta)$ communication rounds. This
leads to the first deterministic $(\Delta+1)$-coloring algorithm in the
standard CONGEST model with a time complexity of $O(\sqrt{\Delta}\cdot polylog
\Delta+\log^* n)$, matching the best time complexity in the LOCAL model up to a
$polylog\Delta$ factor.
