Let $G=(V;E)$ be an infinite graph, with $|V|$ an uncountable cardinal. Suppose that the maximum degree of any $v \in V$ is $\kappa$, where $\kappa \leq |V|$ is an infinite cardinal. Does it follow that the chromatic number of $G$ is at most $\kappa$?
In other words, does Brooks' theorem adapt to infinite graphs?