Waiter–Client triangle-factor game on the edges of the complete graph

Consider the following game played by two players, called Waiter and Client, on edges of K n (where is divisible 3). Initially, all are unclaimed. In each round, picks yet unclaimed edges. Client then chooses one these to be added Waiter’s graph Client’s graph. wins if she forces create a 3 -factor in at some point, while does not manage do that, wins. It difficult see that for large enough , has winning strategy. The question considered Clemens et al. how long will last aims win as soon possible, delay her much both players play optimally. Denote this optimal number rounds τ W C ( F − fac 1 ) . proved 13 12 ≤ 7 6 + o conjectured = note, we verify their conjecture.