In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum degree induced by any color class is at most $\Delta^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded.
翻译:在不适的色彩中, 我们得到一个图形$G$和两个整数$\chi_ d$, $Delta ⁇ $, 并被询问我们是否能够使用 $chi_ d$- color $G$, 使任何彩色类的最大度最多为$\ Delta ⁇ $。 我们显示, 在几个基本图表类中, 彩色的自然概括性要难得多。 特别是, 我们显示, 在分裂的图形中, 即使是两个参数之一 $\ chi_ d$, $\ Delta ⁇ $ 被设定为最小的固定值, 也可能是最小值的固定值。 我们用一个简单的基于树with的 DP 算法来补充这个最小值的固定值 $( $) 。 我们用这个负值来补充这个负值结果, 如果在硬值的彩色图中, 平面值是固定值的, $; 平面的彩度是平面的平面的 。