A balanced schedule exists for 9 rounds where everyone is a bye/substitute exactly once. For example you could play the following:
(2 3 8 6) (4 7 9 5)
(3 1 9 4) (5 8 7 6)
(1 2 7 5) (6 9 8 4)
(5 6 2 9) (7 1 3 8)
(6 4 3 7) (8 2 1 9)
(4 5 1 8) (9 3 2 7)
(8 9 5 3) (1 4 6 2)
(9 7 6 1) (2 5 4 3)
(7 8 4 2) (3 6 5 1)
So each player has 8 games each and plays with the other 8 players exactly 3 times. Hope that helps.
