Consider the gray pixel in the center a placeholder, much like Escher's place holder in the above image (because he couldn't think of how to realistically depict what turns out to be the infinite fractal nature of the center)

> An edge, no matter how small breaks the cross.

My very weak intuition says that an edge of infinite smallness would not.

Now make N as large as you like.

So allowing this makes the problem uninteresting, and precluding it makes the problem interesting and hard.

(I don't recall details - but the preconditions of the Four Colour Theorem rule out all the fuzzy sets, infinitely crooked lines, "this region is all points with one rational and one irrational coordinate", and other trickery that folks who've had a bit of math might otherwise be tempted by.)