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 propertyPisbipartite, in the sense of:

the out-arrows of an

objectnode go to anargumentnode and vice versathe in-arrows of an

objectnode come from anargumentnode and vice versa.