Fine-grained parameterized complexity analysis of graph coloring problems

The $q$-Coloring problem asks whether the vertices of a graph can be properly colored with $q$ colors. Lokshtanov et al. [SODA 2011] showed that on graphs feedback vertex set size $k$ cannot solved in time $\mathcal{O}^*((q-\varepsilon)^k)$, for any $\varepsilon > 0$, unless Strong Exponential-Time Hypothesis (SETH) fails. In this paper we perform fine-grained analysis complexity respect to hierarchy parameters. We show even when parameterized by cover number, must appear base exponent: Unless ETH fails, there is no universal constant $\theta$ such $\mathcal{O}^*(\theta^k)$ all fixed $q$. apply method due Jansen and Kratsch [Inform. & Comput. 2013] prove are $\mathcal{O}^*((q - \varepsilon)^k)$ algorithms where deletion distance several classes $\mathcal{F}$ which known solvable polynomial time. generalize earlier ad-hoc results showing if class whose $(q+1)$-colorable members have bounded treedepth, then exists some 0$ $\mathcal{O}^*((q-\varepsilon)^k)$ given modulator $\mathcal{F}$. contrast, paths simplest unbounded treedepth algorithm exist SETH