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


1. $|C \cap e| = 1$ for all $e\in M$, and
2. $C \subseteq \bigcup M$.


**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?



Is König's Property for graphs inheritable from finite subgraphs?
(Just making my [comment](https://mathoverflow.net/questions/423346/is-k%c3%b6nigs-property-for-graphs-inheritable-from-finite-subgraphs#comment1088018_423346) an answer as [suggested](https://mathoverflow.net/questions/423346/is-k%c3%b6nigs-property-for-graphs-inheritable-from-finite-subgraphs#comment1088048_423346).)


If every finite subgraph of $G$ satisfies Kőnig's Property, then $G$ has no odd cycles and is thus bipartite. Aharoni
([König's Duality Theorem For Infinite Bipartite Graphs](https://doi.org/10.1112/jlms/s2-29.1.1)) proved that if $G$ is bipartite, then $G$ satisfies Kőnig's property.


