( c ) A directed multigraph with distinct vertices U ,...,U3 so that for each i , the vertex u , has in - degree i and out - degree i .

Sure, I understand your question. Here's a possible solution for the directed multigraph with the given properties:

We can create a directed multigraph with three vertices U1, U2, U3, such that each vertex has in-degree i and out-degree i for i in {1, 2, 3}. Here's how we can construct it:

- Start with three vertices U1, U2, U3.
- For each vertex Ui, add i outgoing edges to each of the other vertices, Uj ≠ Ui, for j in {1, 2, 3} and j ≠ i. This ensures that each vertex has out-degree i.
- For each vertex Ui, add i incoming edges from each of the other vertices, Uj ≠ Ui, for j in {1, 2, 3} and j ≠ i. This ensures that each vertex has in-degree i.

With this construction, we have a directed multigraph with three vertices, where each vertex has in-degree i and out-degree i for i in {1, 2, 3}.