I think is $ 5 $. is easy to see that each rook can attack $ 8 $ cells. so the number is not smaller than $ 4 $. suppose that the number is $ 4 $. then each intersection of one line and one column on which are the rooks have to be black and each row and each column must have at most one rook. from these ones we get a contradiction. it's easy to find an example for $ 5 $.

there are a total of 32 white spaces. We prove the min number is 4. Each space can attack 7 cells of its color, and 8 cells of the other color. So we want all the cells to be black to attack the most white cells. We have 8*n<32 for n<4. For N=4 we can illustrate an example. Assuming wlog that the top left square is white and dubbed H1, The rest an be shown as g1,e3,c5,a7.