Finer Tight Bounds for Coloring on Clique-Width

We revisit the complexity of the classical k-Coloring problem parameterized by clique-width. This is a very well-studied problem that becomes highly intractable when the number of colors k is large. However, much less is known on its complexity for small, concrete values of k. In this paper, we completely determine the complexity of k-Coloring parameterized by clique-width for any fixed k, under the SETH. Specifically, we show that for all k >= 3,epsilon>0, k-Coloring cannot be solved in time O^*((2^k-2-epsilon)^{cw}), and give an algorithm running in time O^*((2^k-2)^{cw}). Thus, if the SETH is true, 2^k-2 is the "correct" base of the exponent for every k. Along the way, we also consider the complexity of k-Coloring parameterized by the related parameter modular treewidth (mtw). In this case we show that the "correct" running time, under the SETH, is O^*({k choose floor[k/2]}^{mtw}). If we base our results on a weaker assumption (the ETH), they imply that k-Coloring cannot be solved in time n^{o(cw)}, even on instances with O(log n) colors.