If $G=(V,E)$ is a simple, undirected graph, then $C\subseteq V$ is an *edge cover* if $C\cap e \neq \emptyset$ for all $e\in E$.


Let $G=(V,E) $ be a graph with infinite chromatic number. Is every edge cover $C\subseteq V$ infinite?



Edge covers of graphs with $\chi(G) \geq \aleph_0$
Yes. If $C$ is edge cover of $G$ then if $|C| = n < \infty$ then we can color each vertex of $C$ by its own unique color and color everything else by color $n + 1$ and so chromatic number is finite.


