The speed and threshold of the biased perfect matching and Hamilton cycle games

We consider the biased versions of two popular Maker-Breaker games, namely the perfect matching game and the Hamilton cycle game. We give improved bounds on the threshold bias and the speed, i.e. the number of turns that Maker requires to win, of these games. For the perfect matching game, we show that Maker wins in n2 + o(n) turns when the bias is at most n ln n - f(n)n (ln n)5/4 , for any f going to infinity with n and n sufficiently large (in terms off ). For the Hamilton cycle game, we show that there is a constant C such that for any bias b < n lnn - (lnCn n)3/2 , Maker wins in n + Cn root ln n turns. (c) 2023 Published by Elsevier B.V.