Abstract

In Defective Coloring we are given a graph G and two integers \(\mathrm {\chi _d},\varDelta ^*\) and are asked if we can \(\mathrm {\chi _d}\)-color G so that the maximum degree induced by any color class is at most \(\varDelta ^*\). 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 \(\mathrm {\chi _d},\varDelta ^*\) is set to the smallest possible fixed value that does not trivialize the problem (\(\mathrm {\chi _d}=2\) or \(\varDelta ^*=1\)). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs.

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