4. From an $8 \times 8$ chessboard, 10 squares have been cut out, including both white and black squares. How many $1 \times 2$ dominoes can be ensured to be cut from the remaining part at most?
4. 23 .

Notice that each domino contains one white square and one black square.

Thus, if 9 white squares are cut out, it would be impossible to place more than $32-9=23$ dominoes.
Cut the chessboard as shown in Figure 1.
After 10 squares are missing, at most 10 dominoes are damaged, so there are at least 22 complete dominoes left.
We will now prove that it is possible to add one more domino.
Observe the chain of squares in Figure 1 (starting from the second square from the bottom in the leftmost column $a_{2}$ and moving up along the chain of squares). Since the cut-out squares include both white and black squares, there is a cut-out black square following a cut-out white square somewhere along this chain. If these two cut-out squares are adjacent, it means they only damage one domino, thus leaving at least 23 complete dominoes. In the opposite case, there are uncut squares between them. Therefore, change the cut at this point: immediately after the cut-out white square, cut out the next domino (as shown in Figure 2).

Thus, the number of dominoes can be increased by one, resulting in 23 complete dominoes.