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

TL;DR

This paper establishes the threshold bias in the biased Hamiltonicity Maker-Breaker game on complete graphs, showing Maker's winning strategy when the bias is below approximately n/ln n.

Contribution

It determines the asymptotic critical bias for Maker's winning strategy in the biased Hamiltonicity game on K_n.

Findings

Maker wins if b(n) <= (1-o(1))n/ln n for large n

The critical bias threshold is asymptotically (1+o(1))n/ln n

Provides a precise asymptotic for the game's bias threshold

Abstract

We prove that in the biased 1:b Hamiltonicity Maker-Breaker game, played on the edges of the complete graph K_n, Maker has a winning strategy for b(n)<=(1-o(1))n/ln n, for all large enough n.