Let $G$ be a graph with pairwise disjoint vertex classes $A$, $C$ and $B$, such that there is no edge from a vertex in $A$ to a vertex in $B$ and the cardinality of $C$ equals the vertex connectivity of $G$. Show that there exists a complete matching either from $A$ to $C$ or from $C$ to $A$ (or both).
I am not even sure whether we can begin with the simplification that $G$ is tripartite (since removing edges between vertices in $A$ and $B$ might affect the vertex connectivity). And it seems completely puzzling on how to use the vertex connectivity condition.
Any help appreciated!
If $H$ is the bipartite subgraph consisting of all edges between $A$ and $C$, show that a vertex cover in $H$ smaller than $\min\{|A|,|C|\}$ would also be a vertex cut in $G$. Then apply König's theorem.