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

TL;DR

This work tackles the threshold bias in the $b$-biased Oriented-cycle game on $K_n$, focusing on the largest $b$ for which OMaker can force a directed cycle. It advances the existing safety-based framework by formalizing a safe-digraph invariant built from a Uniform Directed Biclique and $\alpha$-structures, and by introducing a cost-function driven strategy that grows a balance parameter $s$ until $s\ge n-b$. The main contribution is a refined strategy that achieves $t(n,\mathcal{C}) \le 0.7845\,n + O(1)$, leveraging the functions $G_b$ and $g_b$ and the $Q$-cost to select moves that preserve safety while expanding the partition $(A,B)$. This work sharpens our understanding of threshold biases in orientation games and suggests a near-$3n/4$ conjectured lower bound, with avenues for further improvement and refinements of the safety-based approach.

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. Bollobás and Szabó 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 $b \geq 0.7845n + O(1)$.

Figures (1)

• Figure 1: A depiction of a safe digraph digraph for which $A \cup B \neq V$. The thick arrow represents all arcs from $A$ to $B$. The thin arrows represent the decisive arcs of the $\alpha$-structures $D[V\setminus B]$ and $D[V\setminus A]$ respectively.

Theorems & Definitions (29)

• Theorem 1.1
• Definition 2.1
• Proposition 2.2: Proposition 2.5 in CLEMENS201721
• Proposition 2.3: Lemma 2.7 in CLEMENS201721
• Definition 2.4
• Proposition 2.5
• proof
• Proposition 2.6
• proof
• Definition 3.1