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arXiv · 2503.16770

An Improved Upper Bound on the Threshold Bias of the Oriented-cycle Game

Anita Liebenau·Abdallah Saffidine·Jeffrey Yang
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Abstract

We study the $b$-biased Oriented-cycle game where two players, OMaker and OBreaker, take turns directing the edges of $K_n$ (the complete graph on $n$ vertices). In each round, OMaker directs one previously undirected edge followed by OBreaker directing between one and $b$ previously undirected edges. The game ends once all edges have been directed, and OMaker wins if and only if the resulting tournament contains a directed cycle. Bollob\'as and Szab\'o asked the following question: what is the largest value of the bias $b$ for which OMaker has a winning strategy? Ben-Eliezer, Krivelevich and Sudakov proved that OMaker has a winning strategy for $b \leq n/2 - 2$. In the other direction, Clemens and Liebenau proved that OBreaker has a winning strategy for $b \geq 5n/6+2$. Inspired by their approach, we propose a significantly stronger strategy for OBreaker which we prove to be winning for $b \geq 0.7841n + O(1)$.

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Anita Liebenau, Abdallah Saffidine, Jeffrey Yang. 2025-03-21. An Improved Upper Bound on the Threshold Bias of the Oriented-cycle Game. https://arxiv.org/abs/2503.16770

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