What is the proof of this property of an Eulerian digraph?
For every partition of the vertex set of an Eulerian digraph into two parts, A and B, the number of arcs from $A$ to $B$ denoted by $m(A,B)$ is equal to the number of arcs from $B$ to $A$ denoted by $m(B,A)$.
And if possible can someone please tell me where I can find a credible source(book, journal, etc.) that states this as one of the properties of an Eulerian digraph.
Consider an Eulerian cycle. Consider only the edges (in order) that cross between $A$ and $B$.
Note that no two consecutive edges can be from $A$ to $B$ or from $B$ to $A$. Thus, edges from $A$ to $B$ and from $B$ to $A$ must alternate.
Therefore, there are equally many of each, so $m(A, B) = m(B, A)$.