The Maker-Breaker directed triangle game

TL;DR

This work introduces and analyzes the Maker-Breaker directed triangle game on tournaments, focusing on directionally constrained winning sets. It develops a structural and probabilistic framework, including a complete characterization of directed triangles in the parity tournament, a cycle-hopping Maker strategy, and a transfer argument showing Maker’s win on random tournaments via an embedded Pi(7). It further introduces a pregame edge-flip (flip-bias) model and determines the flip-bias threshold $\kappa^*(n)\sim n^2/4$, quantifying how many flips Breaker needs to force a win. Together, these results establish core thresholds and strategies for directionally constrained positional games on both deterministic and random tournament boards, laying groundwork for further study of orientation dynamics in Maker-Breaker games.

Abstract

In this work, we investigate Maker-Breaker directed triangle games -- a directionally constrained variant of the classical Maker-Breaker triangle game. Our board of interest is a tournament, and the winning sets constitute all directed triangles ($3$-cycles) present in the tournament. We begin by studying the Maker-Breaker directed triangle game played on a specially defined tournament called the parity tournament, and we identify the board size threshold to be $n=7$, which is to say that if the size (i.e.\ the number of vertices) of the parity tournament equals $n$, Breaker has a winning strategy for $3\leq n< 7$, while Maker can ensure a win for herself for $n\geq 7$. For the $(1:b)$ biased version of this game, we prove that the bias threshold $b^*(n)$ satisfies $\sqrt{\left(1/12+o(1)\right)\ n}\leq b^{*}(n) \leq\sqrt{\left(8/3+o(1)\right)\ n}$, which matches the order of magnitude (\ $\sqrt{n}$) of the bias threshold for the undirected counterpart of this game. Next, we consider the game on random tournaments $T(n,p)$ with labeled vertices $1,2,\ldots,n$, such that the edge between $i$ and $j$, for each $i<j$, is directed from $i$ towards $j$ with probability $p$, independent of all else. We prove that Maker wins this game with probability approaching $1$ as $n \to \infty$ for any fixed $p \in (0,1)$. Extending the notion of `bias' from undirected games to our directed framework, we introduce the $(1: κ(n))$ flip-biased Maker-Breaker directed triangle game on the parity tournament, where we allow Breaker to strategically flip the directions of a fixed number of edges $κ(n)$ before the game begins. We show that the flip-bias threshold $κ^*(n)$ is asymptotically equal to $n^2/4$ as $n \to \infty$. This work opens up the possibility of studying a variety of directionally constrained Maker-Breaker positional games on tournaments (and more generally, on directed graphs).

Figures (17)

• Figure 1: Example of a directed triangle $(1,2,3)$ in a parity tournament. Here, $1+2=3$ (odd), $2+3=5$ (odd), and $1+3=4$ (even), satisfying the conditions in Subsection \ref{['subsec:structuralcharacterization']}.
• Figure 2: Pairing on $\Pi(5)$'s hypergraph representation
• Figure 3: Diagram of $\Pi(5)$ with edges labeled $a_1,\dots,a_{10}$ and dashed arcs showing Breaker's pairing strategy.
• Figure 4: $\Pi(6)$
• Figure 5: $\Pi(6)$'s hypergraph representation

Theorems & Definitions (40)

• Definition 1.1: Tournament
• Definition 1.2: (1:1) Maker–Breaker directed triangle game
• Definition 1.3: Parity tournament
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.1
• Theorem 2.4
• Lemma 3.1
• proof