Show that a directed multigraph having no isolated vertices has an euler circuit if and only if the graph is weakly connected and the in-degree and out-degree of each vertex are equal

If we let

p as the directed multigraph that has no isolated vertices and has an Euler circuit

q as the graph that is weakly connected with the in-degree and out-degree of each vertex equal

The statement we have to prove is

p ←→q (for biconditional)

Since

p → q (assuming that p is strongly connected to q)

q ← p (since p is strongly connected to q)

Therefore, the bicondition is satisfied