The critical bias for the Hamiltonicity game is (1+𝑜(1))𝑛/ln𝑛

Abstract

We prove that in the biased ( 1 : b ) (1:b) Hamiltonicity Maker-Breaker game, played on the edges of the complete graph K n K_n , Maker has a winning strategy for b ( n ) ≤ ( 1 − 30 ln 1 / 4 ⁡ n ) n ln ⁡ n b(n)\le \left (1-\frac {30}{\ln ^{1/4}n}\right )\frac {n}{\ln n} , for all large enough n n .