Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Despite the fact that some vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view. The equivalence beten \emph{optimization} and \emph{polyhedral separation} suggests that, for these problems, there must exist formulations admitting some elegant characterization for the polytopes associated to them. Therefore, it is interesting to study known formulations for vertex coloring with the goal of finding such characterizations. In this work  study the asymmetric representatives formulation and  show that the corresponding coloring polytope, for a given graph $G$, can be interpreted as the stable set polytope of another graph obtained from $G$. This result allows us to derive complete characterizations for the corresponding coloring polytope for some families of graphs, based on known complete characterizations for the stable set polytope.
['Victor A. Campos', 'Ricardo C. Corrêa', 'Diego Delle Donne', 'Javier Marenco', 'Annegret Katrin Wagler']
I am writing a research paper on a idea Polyhedral studies of vertex coloring problems: The asymmetric representatives formulation
[{"from": "human", "value": "I am writing a research paper on a idea Polyhedral studies of vertex coloring problems: The asymmetric representatives formulation"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Despite the fact that some vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not \"under control\" from a polyhedral point of view. The equivalence beten \\emph{optimization} and \\emph{polyhedral separation} suggests that, for these problems, there must exist formulations admitting some elegant characterization for the polytopes associated to them. Therefore, it is interesting to study known formulations for vertex coloring with the goal of finding such characterizations. In this work  study the asymmetric representatives formulation and  show that the corresponding coloring polytope, for a given graph $G$, can be interpreted as the stable set polytope of another graph obtained from $G$. This result allows us to derive complete characterizations for the corresponding coloring polytope for some families of graphs, based on known complete characterizations for the stable set polytope."}]