Coloring intersection graph of squares

Question

It is known that the coloring intersection graph of axis-parallel rectangles is NP-Hard. What about squares and more specific case "unit squares"?

Thanks.

Answer 1

Even computing a maximum independent set of unit axis-parallel squares is known to be np-hard:

https://www.sciencedirect.com/science/article/pii/0020019081901113?via%3Dihub

Since coloring is a "harder" problem, it should also be NP-hard. A constant approximation follows as if a point is covered by $k$ squares, then the chromatic number is at least $k$, and it is not hard to color such a graph using $O(k)$ colors, observing that the intersection graph has $O(nk)$ edges.