We study biased Maker-Breaker games on the edges of the complete graph, as introduced by Chvatal and Erdos. We show that Maker, occupying one edge in each of his turns, can build a spanning tree, even if Breaker occupies b ≤ (1 - o(1)) · $ {n \over \ln n} $ edges in each turn. This improves a result of Beck, and is asymptotically best possible as witnessed by the Breaker-strategy of Chvatal and Erdos. We also give a strategy for Maker to occupy a graph with minimum degree c (where c = c(n) is a slowly growing function of n) while playing against a Breaker who takes b ≤ (1 - o(1)) · $ {n \over \ln n} $ edges in each turn. This result improves earlier bounds by Krivelevich and Szabo. Both of our results support the surprising random graph intuition: the threshold bias is asymptotically the same for the game played by two “clever” players and the game played by two “random” players. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009
['Heidi Gebauer', 'Tibor Szabó']
Asymptotic random graph intuition for the biased connectivity game