Let $G = (V,E)$ be a simple, undirected graph. A set $C \subseteq V$ is said to be a (vertex) cover if $C \cap e \neq \emptyset$ for all $e\in E$. A matching is a set $M\subseteq E$ of pairwise disjoint edges (elements of $E$).
We say that $G$ has König's Property if there is a matching $M\subseteq E$ and a cover $C\subseteq V$ satisfying
- $|C \cap e| = 1$ for all $e\in M$, and
- $C \subseteq \bigcup M$.
Question. Suppose $G = (V,E)$ is a graph such that for all finite subsets $E_0\subseteq E$ the graph $(V, E_0)$ has König's Property. Does this imply that $G$ has König's Property?