Covering a rectangular chessboard with staircase walks

Let C(n,m)C(n,m) be a n×mn×m chessboard. An ascending (respectively descending) staircase   walk on C(n,m)C(n,m) is a rook’s path on C(n,m)C(n,m) that in every step goes either right or up (respectively right or down). We determine the minimal number of ascending and descending staircase walks covering C(n,m)C(n,m).