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 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.