  Let $G$ be a non-trivial finite group. The well-known Dold's theorem states
that: There is no continuous $G$-equivariant map from an $n$-connected
simplicial $G$-complex to a free simplicial $G$-complex of dimension at most
$n$. In this paper, we give a new generalization of Dold's theorem, by
replacing "dimension 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 Dold's theorem. In
particular, we show that this new parameter is significantly better than the
dimension of target space "for almost all free $\mathbb{Z}_2$-simplicial
complex." In addition, some other applications of strong compatibility graphs
will be presented as well. In particular, a new way for constructing
triangle-free graphs with high chromatic numbers from an n-sphere
$\mathbb{S}^n$, and some new results on the limitations of topological methods
for determining the chromatic number of graphs will be given.

Combinatorics;Algebraic Topology