John,

With 10 players it takes 9 weeks for a player to play everybody else. You can do it like this: place dots 1, 2, ..., 9 equally spaced on a circle and dot 10 in the centre. The schedule for week k consists of the players matched by drawing a straight line segment from 10 to k, and between the players such that the straight line segment between them is perpendicular to the one from 10 to k. Unless I make a typing mistake, this gives you

W1: 10v1, 9v2, 8v3, 7v4, 6v5
W2: 10v2, 1v3, 9v4, 8v5, 7v6
W3: 10v3, 2v4, 1v5, 9v6, 8v7
W4: 10v4, 3v5, 2v6, 1v7, 9v8
W5: 10v5, 4v6, 3v7, 2v8, 1v9
W6: 10v6, 5v7, 4v8, 3v9, 2v1
W7: 10v7, 6v8, 5v9, 4v1, 3v2
W8: 10v8, 7v9, 6v1, 5v2, 4v3
W9: 10v9, 8v1, 7v2, 6v3, 5v4

The same method works for en even number of players. When there are an odd number it is similar but one has to sit out each week.

Have fun.
Victoria