Unidentified Combinatorial Problem

This problem is equivalent to the graph orientation problem also known as the graph balancing problem. One is given an undirected graph and has to give an orientation of the edges which minimizes the maximum out-degree. If this value is $k$, then the graph is called $k$-orientable. Here are some articles on the topic "Graph Orientation Algorithms to Minimize the Maximum Outdegree", and "A note on graph balancing problems with restrictions".

Let me also state an explicit criterion of the existence of orientation with out-degrees at most $d$: any induced subgraph on some, say, $k$ vertices contains at most $dk$ edges. This is clearly necessary, and the proof that it is sufficient is not hard: orient edges arbitrarily and consider the following procedure.

If out-degree of some vertex $a$ is at least $d+1$, then consider the set of vertices $x$, for which there exist oriented path from $a$. If all out-degrees of such vertices are at least $d$, then the set of them contradicts to our assumption. If degree of $x$ is less then $d$, then invert all edges on the path from $a$ to $x$.

Repeating this stuff we kill all high (more then $d$) out-degree after a finite number of steps.