Given a set of straight lines, let each intersection define a vertex. Note that the lines must extend indefinitely, and cannot stop at a vertex (thus forcing every vertex to have even degree).
I aim to prove something along the lines of:
Suppose that there are $n$ sets of parallel lines, each with a distinct gradient. Then every vertex has degree $6$ if and only if there are $3$ sets of equally spaced parallel lines.
Informally, I'm trying to show that there cannot be any configuration of lines such that exactly three lines meet at every intersection, unless the lines "look like" the following figure (only a portion of the lines are shown).
Of course, there is an infinite number of lines in this case.
We can dispose of the case with two sets of parallel lines since no vertex has three lines passing through it. Analysis of small cases doesn't lead me to a method that I can use in general, since (after disposing of the $n=2$ case) I want to prove that if I have more than $3$ distinct gradients, there is always a point of intersection with exactly two lines passing through it.