Rectangle Dissection Into Smaller Rectangles yields shared side

Question

Let $R$ be a rectangle disected into $N$ smaller rectangles with sides parallel to those of $R$. Is there any known condition on N such that if that holds we can always find $2$ rectangles sharing a common edge? (i.e $2$ rectangles able to form a bigger rectangle by merging). Thanks in advance.

Answer 1

The condition is precisely $N<5$.

The pattern for $N=5$ where no rectangles share a common edge can be extended without limit by adding one rectangle at a time to the sides of the large rectangle, as indicated in the image.