Graph theory proofs

Question

I am trying to prove that half of the vertex cover of graph is less than it's matching number. The problem is I don't know how to start and what the solution should be like, please help!

Answer 1

Let $G=(V,E)$ be a graph. Denote by $F\subseteq E$ be a maximal matching of $G$ and by $U$ a minimal vertex cover of $G$. For each edge $e=(u,v)\in E$ we know that at least one of $u$ and $v$ is in $U$, so for each edge we have either one vertex in $U$ or two vertices, thus $|F|\leq 2\cdot |U|$. Hence, making it $|U|\geq \frac{|F|}{2}$.

Note that if our $G$ is a bipartite graph, then by König's theorem, we have an equality between the sizes.