Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: If a graph G contains no subgraph isomorphic to some graph H, then G is called H-free. A coloring of a graph G = (V,E) is a mapping c: V → {1,2,…} such that no two adjacent vertices have the same color, i.e., c(u) ≠ c(v) if uv ∈ E; if |c(V)| ≤ k then c is a k-coloring. The Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k. The Precoloring Extension problem is to decide whether a partial k-coloring of a graph can be extended to a k-coloring of the whole graph for some integer k. The List Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u). By imposing an upper bound l on the size of each L(u)  obtain the l-List Coloring problem.  first classify the Precoloring Extension problem and the l-List Coloring problem for H-free graphs.  then show that 3-List Coloring is NP-complete for n-vertex graphs of minimum degree n − 2, i.e., for complete graphs minus a matching, whereas List Coloring is fixed-parameter tractable for this graph class when parameterized by the number of vertices of degree n − 2. Finally, for a fixed integer k > 0, the List k-Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u) that must be a subset of {1,…,k}.  show that List 4-Coloring is NP-complete for P6-free graphs, where P6 is the path on six vertices. This completes the classification of List k-Coloring for P6-free graphs.
['Petr A. Golovach', 'Daniël Paulusma', 'Jian Song']
I am writing a research paper on a idea Closing Complexity Gaps for Coloring Problems on H-Free Graphs
[{"from": "human", "value": "I am writing a research paper on a idea Closing Complexity Gaps for Coloring Problems on H-Free Graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: If a graph G contains no subgraph isomorphic to some graph H, then G is called H-free. A coloring of a graph G\u2009=\u2009(V,E) is a mapping c: V\u2009\u2192\u2009{1,2,\u2026} such that no two adjacent vertices have the same color, i.e., c(u)\u2009\u2260\u2009c(v) if uv\u2009\u2208\u2009E; if |c(V)|\u2009\u2264\u2009k then c is a k-coloring. The Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k. The Precoloring Extension problem is to decide whether a partial k-coloring of a graph can be extended to a k-coloring of the whole graph for some integer k. The List Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u). By imposing an upper bound l on the size of each L(u)  obtain the l-List Coloring problem.  first classify the Precoloring Extension problem and the l-List Coloring problem for H-free graphs.  then show that 3-List Coloring is NP-complete for n-vertex graphs of minimum degree n\u2009\u2212\u20092, i.e., for complete graphs minus a matching, whereas List Coloring is fixed-parameter tractable for this graph class when parameterized by the number of vertices of degree n\u2009\u2212\u20092. Finally, for a fixed integer k\u2009>\u20090, the List k-Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u) that must be a subset of {1,\u2026,k}.  show that List 4-Coloring is NP-complete for P6-free graphs, where P6 is the path on six vertices. This completes the classification of List k-Coloring for P6-free graphs."}]