Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  define a variant of the H-coloring problem where the number of preimages of certain vertices is predetermined as part of the problem input.  consider the decision and the counting version of the problem; namely the restrictive H-coloring and the restrictive # H- coloring problems.  provide a dichotomy theorem characterizing the H's for which the restrictive H-coloring problem is either NP-complete or polynomially solvable. Moreover,  prove that the same criterion discriminates the #P-complete and the polynomially solvable cases of the restrictive #H-coloring problem. Finally,  prove that both results apply also to the list versions of the above problems.
['Josep Díaz', 'Maria J. Serna', 'Dimitrios M. Thilikos']
I am writing a research paper on a idea The Complexity of Restrictive H-Coloring
[{"from": "human", "value": "I am writing a research paper on a idea The Complexity of Restrictive H-Coloring"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  define a variant of the H-coloring problem where the number of preimages of certain vertices is predetermined as part of the problem input.  consider the decision and the counting version of the problem; namely the restrictive H-coloring and the restrictive # H- coloring problems.  provide a dichotomy theorem characterizing the H's for which the restrictive H-coloring problem is either NP-complete or polynomially solvable. Moreover,  prove that the same criterion discriminates the #P-complete and the polynomially solvable cases of the restrictive #H-coloring problem. Finally,  prove that both results apply also to the list versions of the above problems."}]