Let G be a nontrivial finite group. The wellknown Doldâs theorem states that: There is no continuous Gequivariant map from an connected simplicial Gcomplex to a free simplicial Gcomplex of dimension at most n. In this paper, we give anew generalization of Doldâs theorem, by replacing ââdimension at most n" with a sharper combinatorial parameter. Indeed, this parameter is the chromatic number of a new family of graphs,called strong compatibility graphs,associated to the target space.Moreover, in a series of examples, we will see that one can hope to infer much more information from this generalization than ordinary Doldâs theorem. In particular, we show that this new parameter is significantly better than the dimension of target space ââfor almost all freeZ2simplicial complex." Finally,as another application of strong compatibility graphs, some new results on the limitations of topological methods for determining the chromatic number of graphs will be presented.
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