2-regular directed graphs where the commutative property or relation holds at every vertex and abelian Cayley digraphs.
You are given a 2 regular (2-in 2-out) directed graph where you can check that the commutative relation holds at every vertex. Can you conclude that this is a 2-generated Cayley digraph of an abelian group?. That is, are there 2 regular directed graphs such as the commutative relation holds at every vertex but are not a 2-generated abelian Cayley digraph?.
By commutative relation I mean the usual, as showed in the following image.
Edit. In case the question is not clear I have added an image with examples about what I refer when I say that commutative property holds at every vertex. Also let me clarify that I refer to simple (no loops and no multiple arcs) and connected (one piece) directed graphs. Finally I rephrase the question in an alternative way: is every 2-in 2out directed graph where the commutative property holds at every vertex isomorphic to a 2-generated Abelian Cayley digraph. I mean isomorphic as a digraph (no labels etc...).
