Consider a finite set $X$ of order $n$ and a symmetric function $f: X \times X \rightarrow X$. 

$f$ can  straightforwardly be considered as a [multidigraph][1] with 

 - $n$ "object" nodes, representing the elements of $X$, and 
   
 - $n(n+1)/2$ "argument" nodes, representing the pairs of arguments
   of $f$.

Each of the $n$ object nodes has $n+1$ out-arrows to its corresponding argument nodes. Each of the $n(n+1)/2$ argument nodes has exactly 2 in-arrows from its correspoding object nodes and 1 out-arrow to its corresponding "function value" node (an object node).

Now invert the situation and consider an arbitrary multidigraph with $N = n + n(n+1)/2 = n(n+3)/2$ nodes with the property **P**, that $n$ of them (the *object* nodes) have $n+1$ out-arrows and another $n(n+1)/2$ of them (the *argument* nodes) have exactly 2 in-arrows and 1 out-arrow.

> **Question**: Can - or rather: under which conditions can - be shown that a
> multidigraph with property **P** is
> **bipartite**, in the sense of:
> 
> - the out-arrows of an *object* node go to an *argument* node and vice
> versa
> 
> - the in-arrows of an *object* node come from an *argument* node and vice
> versa.






  [1]: http://en.wikipedia.org/wiki/Multigraph