In this paper, we give list coloring variants of simple iterative defective coloring algorithms. Formally, in a list defective coloring instance, each node $v$ of a graph is given a list $L_v$ of colors and a list of allowed defects $d_v(x)$ for the colors. Each node $v$ needs to be colored with a color $x\in L_v$ such that at most $d_v(x)$ neighbors of $v$ also pick the same color $x$. For a defect parameter $d$, it is known that by making two sweeps in opposite order over the nodes of an edge-oriented graph with maximum outdegree $\beta$, one can compute a coloring with $O(\beta^2/d^2)$ colors such that every node has at most $d$ outneighbors of the same color. We generalize this and show that if all nodes have lists of size $p^2$ and $\forall v:\sum_{x\in L_v}(d_v(x)+1)>p\cdot\beta$, we can make two sweeps of the nodes such that at the end, each node $v$ has chosen a color $x\in L_v$ for which at most $d_v(x)$ outneighbors of $v$ are colored with color $x$. Our algorithm is simpler and computationally significantly more efficient than existing algorithms for similar list defective coloring problems. We show that the above result can in particular be used to obtain an alternative $\tilde{O}(\sqrt{\Delta})+O(\log^* n)$-round algorithm for the $(\Delta+1)$-coloring problem in the CONGEST model. The neighborhood independence $\theta$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. It is known that by doing a single sweep over the nodes of a graph of neighborhood independence $\theta$, one can compute a $d$-defective coloring with $O(\theta\cdot \Delta/d)$ colors. We extend this approach to the list defective coloring setting and use it to obtain an efficient recursive coloring algorithm for graphs of neighborhood independence $\theta$. In particular, if $\theta=O(1)$, we get an $(\log\Delta)^{O(\log\log\Delta)}+O(\log^* n)$-round algorithm.

翻译：本文给出了简单迭代缺陷着色算法的列表着色变体。形式化地，在列表缺陷着色实例中，图的每个节点$v$被赋予一个颜色列表$L_v$以及每个颜色$x$的允许缺陷列表$d_v(x)$。每个节点$v$需选择颜色$x\in L_v$，使得至多有$d_v(x)$个$v$的邻居也选择相同颜色$x$。对于缺陷参数$d$，已知通过对边导向图（最大出度为$\beta$）的节点按相反顺序进行两次扫描，可计算出一种使用$O(\beta^2/d^2)$种颜色的着色，使得每个节点至多有$d$个同色出邻居。我们将该结果推广，证明若所有节点的列表大小为$p^2$且$\forall v:\sum_{x\in L_v}(d_v(x)+1)>p\cdot\beta$，则通过对节点进行两次扫描后，每个节点$v$能选出颜色$x\in L_v$，使得至多有$d_v(x)$个$v$的出邻居被着为颜色$x$。我们的算法比现有同类列表缺陷着色问题的算法更简单，且计算效率显著更高。特别地，我们证明上述结果可用于在CONGEST模型中为$(\Delta+1)$-着色问题提供另一种$\tilde{O}(\sqrt{\Delta})+O(\log^* n)$轮算法。图的邻域独立性$\theta$定义为图中某个节点最多有多少个两两不相邻的邻居。已知通过对邻域独立性为$\theta$的图节点进行单次扫描，可计算出使用$O(\theta\cdot \Delta/d)$种颜色的$d$-缺陷着色。我们将该方法扩展到列表缺陷着色场景，并利用它获得针对邻域独立性为$\theta$的图的高效递归着色算法。特别地，当$\theta=O(1)$时，我们得到$(\log\Delta)^{O(\log\log\Delta)}+O(\log^* n)$轮算法。