Question

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

Answer 1

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