A New Bridge Tournament Schedule

Consider \(n\) bridge teams each consisting of two pairs (the two pairs are called \({teammates}\)). A match is a triple \((i, j, b)\) where pair \(i\) opposes pair \(j\) on a board \(b\); here \(i\) and \(j\) are not teammates and “oppose” is an ordered relation. The problem is to schedule a tournament for the \(n\) teams satisfying the following conditions with a minimum number of boards:

1. Every pair must play against every other pair not on its team exactly once.
2. Every pair must play one match at every round.
3. Every pair must play every board exactly once except for odd \(n\), each pair can skip a board.
4. If pair \(i\) opposes pair \(j\) on a board, then the teammate of \(j\) must oppose the teammate of \(i\) on the same board.
5. Every board is played in at most one match at a round.

We call a schedule satisfying the above five conditions a \({complete \; coupling \; round \; robin \; schedule}\) (CCRRS) and one satisfying the first four conditions a \({coupling \; round \; robin \; schedule}\) (CRRS). While the construction of CCRRS is a difficult combinatorial problem, we construct CRRS for every \(n \geq 2\).