Chef and Surprise Chessboard

Chef loves to play chess, so he bought a new chessboard with width $M$ and height $N$ recently. Chef considers a chessboard *correct* if its width (number of columns) is equal to its height (number of rows) and each cell has no side-adjacent cell of the same color (this is the so-called "chess order" which you can see in real-world chessboards). Chef's chessboard does not have to be a correct che