Checking if a $k$-subset of a graph is a vertex cover in time $O(kn)$

Question

Given a graph $G=(V,E)$ with $|V|=n,|E|=m$. I am reading a $\textit{brute force}$ solution to determining whether each candidate vertex cover of size $k \leq n$ is a vertex cover. The graph does not have loops.

As per page 2 of the notes here we have:

• 1) $C(n,k) = O(n^k)$ $k$-subsets of $V$
• 2) $O(kn)$ time to check whether a subset is a vertex cover

I am considering 2).

What I would think of would be to take each edge in our graph, for which there is a maximum of ${n \choose 2} = O(n^2)$, and check the candidate subset to see if at least one of the vertices is an endpoint of the edge, which would take $O(kn^2)$ checks.

I am not sure how the author arrived at $O(kn)$ operations, any insights appreciated.