I believe that it is not possible to play 5 rounds without having a few pairs play twice.
Here is a four round solution that might help.
( 1 12 9 7) ( 4 10 11) ( 6 3 2) (13 8 5)
( 5 1 10 6) (11 9 8) ( 4 3 7) (12 13 2)
(12 5 3 11) (13 6 9) ( 8 4 1) ( 2 7 10)
( 5 9 2 4) ( 1 3 13) (12 8 10) ( 7 11 6)