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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.

Ong
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1 Answers1

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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 $m(A, B) = m(B, A)$.