proving that there is a directed uv-path in D with indeg(u)=k=outdeg(v) and indegree(v)=outdegree(u)=0

Question

So the question is, suppose u has out-degree k and in-degree 0 while v has indegree k and outdegree 0 where k is a positive integer. And for every other vertex w, the in and out degrees are equal. (a) Prove that there is a directed uv-path in D and (b) Prove that there are k pairwise arc-disjoint directed uv-paths in D.

I am not sure where to begin proving the above questions.. please help!

Answer 1

Hints:

For (a): consider the set $V$ of vertices reachable from $u$ ($u$ not included), and sum their in-degrees into $s_I$, and also their out-degrees into $s_O$. Find an inequality between them, and conclude that we must have $v\in V$.

For (b): same reasoning applies, once you removed an $u-v$ path, so you can do it $k$ times.