Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The $q$-Coloring problem asks whether the vertices of a graph can be properly colored with $q$ colors. Lokshtanov et al. [SODA 2011] shod that $q$-Coloring on graphs with a feedback vertex set of size $k$ cannot be solved in time $\mathcal{O}^*((q-\varepsilon)^k)$, for any $\varepsilon > 0$, unless the Strong Exponential-Time Hypothesis (SETH) fails.   perform a fine-grained analysis of the complexity of $q$-Coloring with respect to a hierarchy of parameters.  show that even when parameterized by the vertex cover number, $q$ must appear in the base of the exponent: Unless ETH fails, there is no universal constant $\theta$ such that $q$-Coloring parameterized by vertex cover can be solved in time $\mathcal{O}^*(\theta^k)$ for all fixed $q$.  apply a method due to Jansen and Kratsch [Inform. & Comput. 2013] to prove that there are $\mathcal{O}^*((q - \varepsilon)^k)$ time algorithms where $k$ is the vertex deletion distance to several graph classes $\mathcal{F}$ for which $q$-Coloring is known to be solvable in polynomial time.  generalize earlier ad-hoc results by showing that if $\mathcal{F}$ is a class of graphs whose $(q+1)$-colorable members have bounded treedepth, then there exists some $\varepsilon > 0$ such that $q$-Coloring can be solved in time $\mathcal{O}^*((q-\varepsilon)^k)$ when parameterized by the size of a given modulator to $\mathcal{F}$. In contrast,  prove that if $\mathcal{F}$ is the class of paths - some of the simplest graphs of unbounded treedepth - then no such algorithm can exist unless SETH fails.
['Lars Jaffke', 'Bart M. P. Jansen']
I am writing a research paper on a idea Fine-Grained Parameterized Complexity Analysis of Graph Coloring Problems
[{"from": "human", "value": "I am writing a research paper on a idea Fine-Grained Parameterized Complexity Analysis of Graph Coloring Problems"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The $q$-Coloring problem asks whether the vertices of a graph can be properly colored with $q$ colors. Lokshtanov et al. [SODA 2011] shod that $q$-Coloring on graphs with a feedback vertex set of size $k$ cannot be solved in time $\\mathcal{O}^*((q-\\varepsilon)^k)$, for any $\\varepsilon > 0$, unless the Strong Exponential-Time Hypothesis (SETH) fails.   perform a fine-grained analysis of the complexity of $q$-Coloring with respect to a hierarchy of parameters.  show that even when parameterized by the vertex cover number, $q$ must appear in the base of the exponent: Unless ETH fails, there is no universal constant $\\theta$ such that $q$-Coloring parameterized by vertex cover can be solved in time $\\mathcal{O}^*(\\theta^k)$ for all fixed $q$.  apply a method due to Jansen and Kratsch [Inform. & Comput. 2013] to prove that there are $\\mathcal{O}^*((q - \\varepsilon)^k)$ time algorithms where $k$ is the vertex deletion distance to several graph classes $\\mathcal{F}$ for which $q$-Coloring is known to be solvable in polynomial time.  generalize earlier ad-hoc results by showing that if $\\mathcal{F}$ is a class of graphs whose $(q+1)$-colorable members have bounded treedepth, then there exists some $\\varepsilon > 0$ such that $q$-Coloring can be solved in time $\\mathcal{O}^*((q-\\varepsilon)^k)$ when parameterized by the size of a given modulator to $\\mathcal{F}$. In contrast,  prove that if $\\mathcal{F}$ is the class of paths - some of the simplest graphs of unbounded treedepth - then no such algorithm can exist unless SETH fails."}]