Let $G$ be a connected graph of even order( greater than or equal to 2k) such that every set of $k-1$ independent edges belong to a $1-factor$ of the graph. Then the graph is $k$-connected.
If the fact that $G$ is connected not assumed then the condition that every set of $k$ independent edges belong to a $1-factor$ implies every set of $k-1$ independent edges also belong to a $1-factor$ of $G$ provided the graph has order greater than or equal to 2k+2.