# Defective Coloring on Classes of Perfect Graphs

## TL;DR

This paper investigates the computational complexity of defective coloring across various graph classes, revealing NP-hardness results even on simple classes and providing algorithms for special cases.

## Contribution

It establishes NP-hardness of defective coloring on split graphs and cographs, and provides algorithms for fixed parameters and special graph classes.

## Key findings

- NP-hard on split graphs with fixed parameters
- NP-hard on cographs, a natural class
- Polynomial algorithms for fixed parameters and certain classes

## Abstract

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.