Hi,
If $k$ is a constant and the algorithm depends linearly on $|V|$ and $|E|$, but is independent of $k$, the complexity of your algorithm is $O(|V|+|E|)$. If your algorithm depends on, say, $2^k$, your running time is still $O(|V|+|E|)$, but clearly they depend on k in a different way - if they both take a minute to run when$k=2$, then when $k=22$ the first will still take a minute, but the second could take more than a year.
In the first part, the complexity of your algorithm is allowed to be dependent on k (btw - this is a hint - if you find that it doesn't depend on k, you should probably double-check your solution). In the second part, it is not allowed to depend on k- the algorithm should take the same time to run if k is 2 or 100000.