# Show that a directed multi graph having no isolated vertices has an Euler Circuit iff …

I am trying to prove the following statement:

Show that a directed multi-graph having no isolated vertices has an Euler circuit iff the graph is weakly connected and $\text{deg}^{\text{in}} = \text{deg}^{\text{out}}$ for each vertex.

The problem I am having is understanding how to reduce the statement into two parts because of the "if and only if".

I came up with the following things I have to prove:

1. If the graph has an Euler circuit then it is weakly connected
2. If the graph has an Euler circuit then each vertex must have $\text{deg}^{\text{in}} = \text{deg}^{\text{out}}$

Is this correct? How does one reduce statements of the form $C$ iff $A$ and $B$?

1. If the multigraph (with no isolated vertices) has an Euler Circuit then the graph is weakly connected and $\text{deg}^{\text{in}} = \text{deg}^{\text{out}}$
2. If the graph is weakly connected and $\text{deg}^{\text{in}} = \text{deg}^{\text{out}}$ then the multigraph (with no isolated vertices) has an Euler Circuit?