Sharp thresholds for half-random games II

Jonas Groschwitz*, Tibor Szabó

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this work we focus on the case of Breaker playing randomly and Maker being "clever". The reverse scenario is treated in a separate paper. We determine the sharp threshold bias of classical games played on the edge set of the complete graph Kn, such as connectivity, perfect matching, Hamiltonicity, and minimum degree-1 and -2. In all of these games, the threshold is equal to the trivial upper bound implied by the number of edges needed for Maker to occupy a winning set. Moreover, we show that CleverMaker can not only win against asymptotically optimal bias, but can do so very fast, wasting only logarithmically many moves (while the winning set sizes are linear in n).

Original languageEnglish
Pages (from-to)387-401
Number of pages15
JournalGraphs and Combinatorics
Volume33
Issue number2
DOIs
Publication statusPublished - 2017

Keywords

  • Positional games
  • Randomized strategy
  • Sharp threshold
  • Fast win
  • Hamiltonicity
  • Connectivity

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