We consider the bimodal language, where the first modality is interpreted by a binary relation in the standard way, and the second is interpreted by the relation of inequality. It follows from Hughes (1990), that in this language, non-$k$-colorability of a graph is expressible for every finite $k$. We show that modal logics of classes of non-$k$-colorable graphs (directed or non-directed), and some of their extensions, are decidable.

翻译：我们考虑双模态语言，其中第一个模态以标准方式通过二元关系进行解释，第二个模态通过不等关系进行解释。由Hughes（1990）可知，在此语言中，对于每个有限$k$，图的非$k$-可着色性均可表达。我们证明，非$k$-可着色图类（有向或无向）的模态逻辑及其某些扩展是可判定的。